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Chapter XII.
Physical Applications.

388. We propose to conclude the work by giving a few in stances of the ready applicability of quaternions to questions of mathematical physics, upon which, even more than on the Geometrical or Kinematical applications, the real usefulness of the Calculus must mainly depend—except, of course, in the eyes of that section of mathematicians for whom Transversals and Anharmonic Pencils, &c. have a to us incomprehensible charm. Of course we cannot attempt to give examples in all branches of physics, nor even to carry very far our investigations in any one branch : this Chapter is not intended to teach Physics, but merely to shew by a few examples how expressly and naturally quaternions seem to be fitted for attacking the problems it presents.

We commence with a few general theorems in Dynamics—the formation of the equations of equilibrium and motion of a rigid system, some properties of the central axis, and the motion of a solid about its centre of inertia. The student may profitably compare, with the processes in the text, those adopted by Hamilton in his Elements (Book III., Chap. III., Section 8).

A. Statics of a Rigid System.

389. When any forces act on a rigid body, the force β at the point whose vector is α, &c., then, if the body be slightly displaced, so that a becomes α + δα, the whole work done against the forces is

ΣSβδα.
310 This must vanish if the forces are such as to maintain equilibrium. Hence the condition of equilibrium of a rigid body is
ΣSβδα = 0.
For a displacement of translation δα is any constant vector, hence
  Σβ = 0
 
 (1).
For a rotation-displacement, we have by § 371, ε being the axis, and Tε being indefinitely small,
δα = Vεα,
and ΣS.βVεα = ΣS.εVαβ = S.εΣ(Vαβ) = 0,
whatever be ε, hence
  Σ . Vαβ = 0
 
 (2).
These equations, (1) and (2), are equivalent to the ordinary six equations of equilibrium.

390. In general, for any set of forces, let

Σ β = β1,
Σ . Vαβ = α1,

it is required to find the points for which the couple at has its axis coincident with the resultant force β1. Let γ be the vector of such a point.

Then for it the axis of the couple is

Σ . V(α − γ)β = α1 − Vγβ1,
and by condition xβ1 = α1 − Vγβ1.
Operate by Sβ1 ; therefore
xβ12 = Sα1β1,
and Vγβ1 = α1 − Sα1β1 = −β1Vα1β1−1,
or γ = Vα1β1−1 + yβ1

a straight line (the Central Axis) parallel to the resultant force.

[If the resultant force and couple be replaced by an equivalent in the form of two forces, β at α, and β′ at α′, we have

β + β′ = β1, Vαβ + Vα′β′ = α1.
The volume of the tetrahedron whose opposite edges are β, β′ (acting as above stated) is as S . β′V(α − α′)β. But
S.β′αβ = Sβ′α1, S.β′α′β = −Sβ′α1,

so that the volume is as Sα1(β + β′) = Sα1β1 a constant whatever pair of equivalent forces be taken.]

sections 391-etc. not completed
311

391. To find the points about which the couple is least. Here T (ax - Fy^) = minimum. Therefore 8 . (a, - Fy/^) VPrf = 0, where 7 is any vector whatever. It is useless to try 7 = ftv but we may put it in succession equal to OL I and to Vaft^ Thus and ( FaA)2 - {3*S . 7FA = 0. Hence 7 = x VOLft\ -f yfiv and by operating with S . Va.J3v we get or 7 = 1/ the same locus as in last section.

392. The couple vanishes if This necessitates $,, = 0, or the force must be in the plane of the couple. If this be the case, 7 = 1/3," + */3I still the central axis. To assign the values of forces f, ft , to act at e, ep and be equivalent to the given system. Hence Fef + Fe, (/9, - f) = a,, and f = (6 - e,)- 1 (a, - F^/3,) + x (e - 6 t ). Similarly for fr The indefinite terms may be omitted, as they must evidently be equal and opposite. In fact the}^ are any equal and opposite forces whatever acting in the line joining the given points.

393. If a system of parallel forces act on a rigid body, say x/3 at a, &c. they have the single resultant /3S (a), at a, such that 312 Hence, whatever be the common direction of the forces, the resultant passes through _ 2 pa)~ If 5 (#) = 0, the resultant is simply the couple

By the help of these expressions for systems of parallel forces we can easily proceed to the case of forces generally.

Thus if any system of forces, /3, act at points, a, of a rigid body ; and if i, j, k be a system of rectangular unit vectors such that the resultant force is , bk acting at , or as we may write it. Take this as origin, then = 0.

The resultant couple, in the same way, is or V(i(f)i+j(f)j). Now i, $7, A? are invariants, in the sense that they retain the same values however the forces and (with them) the system i, j, k be made to rotate : provided they preserve their mutual inclinations, and the forces their points of application. For the as are constant, and quantities of the form S/3i, S/3jt or S/3k are not altered by the rotation.

We may select the positions of i and j so that (fi and fyj shall be perpendicular to one another. For this requires only 8 . ty fj = 0, or S.ipthj = 0. But (§ 381) / (/ is a self-conjugate function; and, by our change of origin, k is parallel to one of its chief vectors. The desired result is secured if we take i, j as the two others.

With these preliminaries we may easily prove Minding’s Theorem :

If a system of forces, applied at given points of a rigid body, have their directions changed in any way consistent with the preservation of their mutual inclinations, they have in an infinite number of positions a single force as resultant. The lines of action of all such single forces intersect each of two curves fixed in space.

313 The condition for the resultant’s being a single force in the line whose vector is p is bVkp=V(i(j)i+j(fj\ which may be written as bp = xk J4i + if)j. That the two last terms, together, form a vector is seen by operating on the former equation by S.k , for we thus have We may write these equations for convenience as bp = xk jo. + i/3 ....................... ,(1), Sja-Sij3 = () ....................... (2).

[The student must carefully observe that a and /3 are now used in a sense totally different from that in which they first appeared, but for which they are no longer required. If this should puzzle him, he may change a into 7, and (3 into 8, in the last two equations and throughout the remainder of this section.]

Our object now must be to express i and j in terms of the single variable k, which is afterwards to be eliminated for the final result.

From (2) we find at once whence we easily arrive at either of the following or _ y = a* + + (Ska)* + (Sk/3) z + 28 . kcfi] Substituting for i and^ in (1) their values (3), we have = (vr-z)k-a/3 ........................... (5), where wt which is now used for a linear and vector function, is defined by the equation srp = aSap ftSffp. Obviously r (a/3) = 0, so that (w - z)~ l (a/3) = - - a/3. z Thus -yb(^-2)-l p = k+-..................... (6). 314 Multiply together the respective members of (5) and (6), and take the scalar, and we have or, by (4), = f + z + a* + /3 2 + z = /+ z which, for z a2 , or z = - /3 2 , gives as the required curves the focal conies of the system

394. The preceding investigation was based on the properties of a system of parallel forces ; and thus has a somewhat composite, semi-Cartesian, character.

That which follows is much more purely quaternionic. It is taken from the Trans. R. S. E. 1880.

When any number of forces act on a rigid system ; (3l at the point GLV /32 at 2, &c., their resultant consists of the single force /8 = 2/3 acting at the origin, and the couple * = -2F/3a........................... (1).

If these can be reduced to a single force, the equation of the line in which that force acts is evidently F/8p = 2F/3a .............................. (2).

Now suppose the system of forces to turn about, preserving their magnitudes, their points of application, and their mutual inclinations, and let us find the fixed curves in space, each of which is intersected by the line (2) in every one of the infinite number of its positions.

Operating on (2) by F. /3, it becomes with the notation of Chap. V. Now, however the forces may turn, 0/3 = 2a/3/3 is an absolute constant ; for each scalar factor as S/3J3 is unaltered by rotation. Let us therefore change the origin, i.e. the value of each a, so as to make = ........................ (3).

315 This shews that / is one of the three principal vectors of $, and we see in consequence that $ may be expressed in the form where 7 and S are unit vectors, forming with JB a rectangular system. They may obviously be so chosen that 7 and 8 shall be at right angles to one another, but these (though constants) are not necessarily unit vectors.

Equation (2) is now F/3P =F77 +FSS ..................... (2 ), where b is the tensor, and ft the versor, of ft. The condition that the force shall lie in the plane of the couple is, of course, included in this, and is found by operating by S .0. Thus 8 (/ - 78 ) = ........................... (4).

395. We have here all the data of the problem, and solutions can only differ from one another in the mode of attacking and (4).

Writing (4) in the form 7 (8 we have at once ty = F/3S + /3 Vfiy, } ,.,. whence tS = - V@y + ft V/3 j" where t is an undetermined scalar.

By means of these we may put (2 ) in the form where *f = -ySy ( )-S SS ( ). Let the tensors of 7 and 8" be ev e2 respectively, and let be a unit vector perpendicular to them, then we may write btp^xp-e.efl + isp ........................ (5). Operating by (-or + a;)" 1 , and noting that we have bt (cr + = /3 - -- ................. (5 ). M? Taking the scalar of the product of (5) and (5 ) we have + x)~ l p = - (sc/3 - ep

316 But by (4 ) we have (6), so that, finally, (7).

Equation (7), in which f is given by (6) in terms of ft, is true for every point of every single resultant. But we get an immense simplification by assuming for x either of the particular values e* or e*. For then the right-hand side of (7) is reduced to unity, and the equation represents one or other of the focal conies of the system of confocal surfaces a point of each of which must therefore lie on the line (5).

396. A singular form, in which Minding’s Theorem can be expressed, appears at once from equation (2 ). For that equation is obviously the condition that the linear and vector function -bpS0( ) + y8y( ) + m( ) shall denote a pure strain.

Hence the following problem : Given a set of rectangular unit vectors, which may take any initial position : let two of them, after a homogeneous strain, become given vectors at right angles to one another, find what the third must become that the strain may be pure. The locus of the extremity of the third is, for every initial position, one of the single resultants of Minding’s system ; and therefore passes through each of the fixed conies.

Thus we see another very remarkable analogy between strains and couples, which is in fact suggested at once by the general expression for the impure part of a linear and vector function.

397. The scalar t, which was introduced in equations (4 ), is shewn by (6) to be a function of ft alone. In this connection it is interesting to study the surface of the fourth degree - (e* + e*) r2 - 2e 1 e 22Wr= 1, where r = - ft. L But this may be left as an exercise. Another form of t (by 4 ) is $77 + SSB .

317 Meanwhile (6) shews that for any assumed value of ft there are but two corresponding Minding lines. If, on the other hand, p be given there are in general four values of /3.

398. For variety, and with the view of further exploring this very interesting question, we may take a different mode of attacking equations (4) and (2 ), which contain the whole matter. In what follows b will be merged in p, so that the scale of the result will be altered.

Operating by V. ft we transform (2 ) into p + pS0p=-(ySyi3 + 8SVl3) (2"). Squaring both sides we have p* + S*/3p = S/3/3 (8). Since /3 is a unit vector, this may be taken as the equation of a cyclic cone; and every central axis through the point p lies upon it. For we have not yet taken account of (4), which is the condition that there shall be no couple.

To introduce (4), operate on (2") by S.y and by S.8 . We thus have, by a double employment of (4), Next, multiplying (8) by Sftvrfi, and adding to it the squares of (9), we have p*Sfa/3 - 2S0pSfap - Spvp = - Sfa P (10). This is a second cyclic cone, intersecting (8) in the four directions /3. Of course it is obvious that (8) and (10) are unaltered by the substitution of p + y/3 for p.

If we look on ft as given, while p is to be found, (8) is the equation of a right cylinder, and (10) that of a central surface of the second degree.

399. A curious transformation of these equations may be made by assuming p1 to be any other point on one of the Minding lines represented by (8) and (10). Introducing the factor -/32 ( = 1) in the terms where @ does not appear, and then putting throughout II Pi-P (11) (8) becomes - p 2 Pl 2 + S*pp1 = S (Pl - p) (Pl - p) (8 ). As this is symmetrical in p, pv we should obtain only the same

318 result by putting pl for p in (8), and substituting again for /3 as before.

From (10) we obtain the corresponding symmetrical result (p 2 - SppJ Spl^pl + (p, 2 - SppJ Spvp = - Sppfi (p, - p) VF (pl - p) -S(pi -p)^(Pl -p) ...... (10 ). These equations become very much simplified if we assume p and pl to lie respectively in any two conjugate planes ; specially in the planes of the focal conies, so that SS p = 0, and Syf pl = 0.

For if the planes be conjugate we have i = 0, and if, besides, they be those of the focal conies, Bph SffpBffto 8pvr 2 p = e*Spwp, &c., and the equations are P 2 p 2 and p*Splpl + p*Spwp = 8p1 vr 3 pl Spiv 2 p ......... (1 0"). From these we have at once the equations of the two Minding curves in a variety of different ways. Thus, for instance, let pt =p* and eliminate p between the equations. We get the focal conic in the plane of ft , 7 . In this way we see that Minding lines pass through each point of each of the two curves ; and by a similar process that every line joining two points, one on the one curve, the other on the other, is a Minding line.

400. Another process is more instructive. Note that, by the equations of condition above, we have Then our equations become + Shmpt = 0 and (p 2 + e 2 ) Sp^p, + (p* + e 2 ) Spvp = 0. If we eliminate p 2 or p 2 from these equations, the resultant obviously becomes divisible by Spivp or Sp^pv and we at once obtain the equation of one of the focal conies.

319

401. In passing it may be well to notice that equation (10) may be written in the simpler form 8 . p/3pvr/3 + Sp-vp = Sj3n*l3. Also it is easy to see that if we put we have (8) in the form S/36 = 0, and by the help of this (10) becomes This gives another elegant mode of attacking the problem.

402. Another valuable transformation of (2") is obtained by considering the linear and vector function, x suppose, by which fi, 7, S are derived from the system ft , Uy, US . For then we have obviously P = *XP + X**XP..................... (2 ")- This represents any central axis, and the corresponding form of the Minding condition is S.Jxvr-*V = S.xSF ~*y .................. (4"). Most of the preceding formulae may be looked upon as results of the elimination of the function x from these equations. This forms probably the most important feature of such investigations, so far at least as the quaternion calculus is concerned.

403. It is evident from (2 ") that the vector-perpendicular from the origin on the central axis parallel to xfi is expressed by But there is an infinite number of values of % for which Ur is a given versor. Hence the problem ; to find the maximum and minimum values of Tr, when Ur is given i.e. to find the surface bounding the region which is filled with the feet ofperpendiculars on central axes.

We have 2V = - S . = TrS.x/3 Ur. Hence = S . But as Tj3 is constant = 8. 320 These three equations give at sight where u, u are unknown scalars. Operate by S . %/3 and we have - T2r-u=0, so that ST (sr + r2 ) 1 r = 0.

This differs from the equation of Fresnel’s wave-surface only in having w + r2 instead of TZ + r~2 , and denotes therefore the reciprocal of that surface. In the statical problem, however, we have izft = 0, and thus the corresponding wave-surface has zero for one of its parameters. [See § 435.]

[If this restriction be not imposed, the locus of the point where (/ is now any given linear and vector function whatever, will be found, by a process precisely similar to that just given, to be 8.(r- f ) (f + T")- (T - f/3 ) = 0, where * is the conjugate of /.]

B. Kinetics of a Rigid System.

404. For the motion of a rigid system, we have of course by the general equation of Lagrange.

Suppose the displacements a to correspond to a mere translation, then So. is any constant vector, hence 2 (ma - /3) = 0, or, if ax be the vector of the centre of inertia, and therefore we have at once a^m 2/3 = 0, and the centre of inertia moves as if the whole mass were concentrated in it, and acted upon by all the applied forces.

405. Again, let the displacements Sa correspond to a rotation about an axis e, passing through the origin, then it being assumed that Te is indefinitely small.

321 Hence 28 . e Vet (ma - ) = 0, for all values of e, and therefore 2 . Fa (ma - /9) = 0, which contains the three remaining ordinary equations of motion.

Transfer the origin to the centre of inertia, i.e. put a = ^ + CT, then our equation becomes 2F(a, + tsr) (ma, + WOT - ft) = ; or, since 2mar = 0, 2 Far (m# - ) + Fa, (a\2m - 2y3) = 0.

But (§ 404) 3,2m - 2)9 = 0, hence our equation is simply

Now 2 Fsr/3 is the couple, about the centre of inertia, produced by the applied forces ; call it f, then ?........................... (1).

406. Integrating once, 2roFM!r = 7 + /fft ..................... (2).

Again, as the motion considered is relative to the centre of inertia, it must be of the nature of rotation about some axis, in general variable. Let e denote at once the direction of, and the angular velocity about, this axis. Then, evidently, CT = Fe-sr.

Hence, the last equation may be written 2mcr Fear = 7 + fdt. Operating by S . e, we get 2m(Vei*y = Sev + Sft;dt .................. (3).

But, by operating directly by 2fSedt upon the equation (1), we get 2m(FS7)2 = -/^2 + 2/Sfe^ ............... (4). Equations (2) and (4) contain the usual four integrals of the first order, h being here an arbitrary constant, whose value depends upon the initial kinetic energy of the system. By § 387 we see how the principal moments of inertia are involved in the left member. T. Q. I. 21

322 407. When no forces act on the body, we have f=0, and therefore Sm-crFe sr = y ........................ (5), 2mr8 = 2m(F6w)8 = -^ .................. (6), and, from (5) and (6), Se7 = -/i2 ........................... (7). One interpretation of (6) is, that the kinetic energy of rotation remains unchanged : another is, that the vector e terminates in an ellipsoid whose centre is the origin, and which therefore assigns the angular velocity when the direction of the axis is given ; (7) shews that the extremity of the instantaneous axis is always in a plane fixed in space.

Also, by (5), (7) is the equation of the tangent plane to (6) at the extremity of the vector e. Hence the ellipsoid (6) rolls on the plane (7).

From (5) and (6), we have at once, as an equation which e must satisfy, 7 22 . m (Fe^r) 2 = - h* (2 . mw Fesr) 2 . This belongs to a cone of the second degree fixed in the body. Thus all the results of Poinsot regarding the motion of a rigid body under the action of no forces, the centre of inertia being fixed, are deduced almost intuitively: and the only difficulties to be met with in more complex properties of such motion are those of integration, which are inherent to the subject, and appear whatever analytical method is employed. (Hamilton, Proc. R. /. A. 1848.)

If we write (5) as +" 1 6 = 7 ........................... (5), the special notation } indicating that this linear and vector function is related to the principal axes of the body, and not to lines fixed in space, and consider the ellipsoid (of which e is a sernidiameter) Se+^e^-h* ........................ (6), we may write the equation of a confocal ellipsoid (also fixed in the body) as Any tangent plane to this is S.r(4, + Pr P = -h ..................... (9). 323 If this plane be perpendicular to 7, we may write so that, by (5) (10). The plane (9) intercepts on 7 a quantity h*/xTy, which is constant by (8) and (10).

The vector velocity of the point p is Vep px Vey = p Vyp (by two applications of (10)). Hence the point of contact, p, revolves about 7 with angular velocity pTy. That is, if the plane (9) be rough, and can turn about 7 as an axis, the ellipsoid (8) instead of sliding upon it, will make it rotate with uniform angular velocity. This is a very simple mode of obtaining one of Sylvester’s remarkable results. (Phil. Trans. 1866.)

408. For a more formal treatment of the problem of the rotation of a rigid body, we may proceed as follows :

Let ct be the initial position of r, q the quaternion by which the body can be at one step transferred from its initial position to its position at time t. Then w = qctq 1 and Hamilton’s equation (5) of last section becomes S . mqaq" 1 V. eqaq~ l = 7, or S . mq {aS . aq~ l eq q~ l eqc?} q~ l = 7.

The vector 7 is now written for 7 + f%dt of § 406, as f is required for a new purpose. Thus 7 represents the resultant moment of momentum, and will be constant only if there is no applied couple. Let (fp = S m (aSap 2 /o) .................. (1), where $ (compare § 387) is a self-conjugate linear and vector function, whose constituent vectors are fixed in the body in its initial position. Then the previous equation may be written or For simplicity let us write T (1 = q~ l yq = Then Hamilton’s dynamical equation becomes simply 0i? = ? (3). 21-2

324 409. It is easy to see what the new vectors rj and f represent. For we may write (2) in the form from which it is obvious that 77 is that vector in the initial position of the body which, at time t, becomes the instantaneous axis in the moving body. When no forces act, 7 is constant, and f is the initial position of the vector which, at time t, is perpendicular to the invariable plane.

410. The complete statement of the problem is contained in equations (2), (3) above, and (4) of § 372*. Writing them again, we have 7? = Z?.............................. (2), *-? = ? .............................. (3)-

We have only to eliminate f and 77, and we get 27=/r(r 7?) ........................ (-5), in which q is now the only unknown; 7, if variable, being supposed given in terms of q and t.

It is hardly conceivable that any simpler, or more easily interpretable, expression for the motion of a rigid body can be presented until symbols are devised far more comprehensive in their meaning than any we yet have.

411. Before entering into considerations as to the integration of this equation, we may investigate some other consequences of the group of equations in § 410. Thus, for instance, differentiating (2), we have and, eliminating q by means of (4), * To these it is unnecessary to add Tq = constant, as this constancy of Tq is proved by the form of (4). For, had Tq been variable, there must have been a quaternion in the place of the vector 17. In fact, l 325 whence, eliminating y by the help of (2), which gives, in the case when no forces act, the forms = W?.............................. (0). and (as f =(/??) (/7) = -F.77c/?7 ........................ (7). To each of these the term q~lfyq must be added on the right, if forces act,

412. It is now desirable to examine the formation of the function c/. By its definition § 408, (1), we have frp = 2 . m (aSap - a. 2 p), = S. maVap. Hence - Spjp = 2.m (TVap}\ so that Spj)p is the moment of inertia of the body about the vector p, multiplied by the square of the tensor of p. Compare

387. Thus the equation evidently belongs to an ellipsoid, of which the radii-vectores are inversely as the square roots of the moments of inertia about them; so that, if i, j, k be taken as unit-vectors in the directions of its axes respectively, we have (8), A, B, G, being the principal moments of inertia. Consequently _j_p = -{AiSip + BjSjp + CkSkp] ..................... (9). Thus the equation (7) for 77 breaks up, if we put it] = ia) l -\-j(o2 + o)3 , into the three following scalar equations which are the same as those of Euler. Only, it is to be understood that the equations just written are not primarily to be considered as equations of rotation. They rather express, with reference to 326 fixed axes in the initial position of the body, the motion of the extremity, a) lt o 2 , 3 , of the vector corresponding to the instantaneous axis in the moving body. If, however, we consider c^, o^, o 3 as standing for their values in terms of wt x, y, z (§ 416 below), or any other coordinates employed to refer the body to fixed axes, they are the equations of motion.

Similar remarks apply to the equation which determines , for if we put (6) may be reduced to three scalar equations of the form i_i\ in I 23

413. Euler’s equations in their usual form are easily deduced from what precedes. For, let whatever be p ; that is, let } represent with reference to the moving principal axes what * represents with reference to the principal axes in the initial position of the body, and we have c|e = -V. which is the required expression.

But perhaps the simplest mode of obtaining this equation is to start with Hamilton’s unintegrated equation, which for the case of no forces is simply S . mVvriir = 0. But from TS we deduce OT = -sre 2 eSetz + so that S . m ( Vevr Set? etrr 2 + TVSet?) = 0. If we look at equation (1), and remember that _j_ differs from _/_ simply in having trr substituted for a, we see that this may bo written _j_e = 0, 327 the equation before obtained. The first mode of arriving at it has been given because it leads to an interesting set of transformations, for which reason we append other two.

By (2) 7 = qq \ therefore = qq~\ qgq 1 + qgq 1 - qq~l qq~ l , or ^ But, by the beginning of this section, and by (5) of § 407, this is again the equation lately proved.

Perhaps, however, the following is neater. It occurs in Hamilton’s Elements.

By (5) of § 407 |e = 7. Hence {e = je = 2 . m ( = V. e^L .

414. However they are obtained, such equations as those of § 412 were shewn long ago by Euler to be integrable as follows,

Putting Zfco^^co^dt = s, we have Aco* = AQ* + (B-C) s, with other two equations of the same form. Hence 9/7*= ^f . C-A \*/^ A-B so that t is known in terms of s by an elliptic integral. Thus, finally, rj or f may be expressed in terms of t ; and in some of the succeeding investigations for q we shall suppose this to have been done. It is with this integration, or an equivalent one, that most writers on the farther development of the subject have commenced their investigations.

415. By § 406, 7 is evidently the vector moment of momentum of the rigid body ; and the kinetic energy is - 2 . mtsr* = - But #67 = S . ql eqq~ ly 328 so that when no forces act But, by (2), we have also T=T% or so that we have, for the equations of the cones described in the initial position of the body by rj and f, that is, for the cones de scribed in the moving body by the instantaneous axis and by the perpendicular to the invariable plane, This is on the supposition that 7 and h are constants. If forces act, these quantities are functions of t, and the equations of the cones then described in the body must be found by eliminating t between the respective equations. The final results to which such a process will lead must, of course, depend entirely upon the way in which t is involved in these equations, and therefore no general statement on the subject can be made.

416. Recurring to our equations for the determination of q, and taking first the case of no forces, we see that, if we assume rj to have been found (as in § 414) by means of elliptic integrals, we have to solve the equation * To get an idea of the nature of this equation, let us integrate it on the suppo sition that ?j is a constant vector. By differentiation and substitution, we get Hence q = Ql cos - t + Q.2 sin ~- t. Substituting in the given equation we have T-n (-QlBin ^t + Q.2 cos^t^ = (QlCos T J t + Q.sm^ t) r,. Hence ^ . Q2 = Q^, -T*.Qi=Qrfl, which are virtually the same equation, and thus tTr, )* - And the interpretation of q ( ) q~l will obviously then be a rotation about tj through the angle tT-rj, together with any other arbitrary rotation whatever. Thus any position whatever may be taken as the initial one of the body, and Q1 ( ) Q^1 brings it to its required position at time i = 0. 329 that is, we have to integrate a system of four other differential equations harder than the first,

Putting, as in § 412, where o lt co l2) w3 are supposed to be known functions of t, and q = w + ix +jy + kz, 1 ,. dw dx dii dz this system is 9 W = ~X = ~Y = ~7 y where W = w^o w^y /, X ft)jW + DJ ft)./, Y = Z = or, as suggested by Cayley to bring out the skew symmetry, X = . ft) 37/ ft)/ + WVW, Y = ft 3 . + ft)/ + (02W, Z = ft) 2 wjj . 4- o) 3w, W = a)^ a) 2y - )/ Here, of course, one integral is w2 + xz + y* + z2 = constant.

It may suffice thus to have alluded to a possible mode of solution, which, except for very simple values of rj, involves very great difficulties. The quaternion solution, when TJ is of constant length and revolves uniformly in a right cone, will be given later.

417. If, on the other hand, we eliminate TJ, we have to integrate q^(q-l yq) = 2q, so that one integration theoretically suffices. But, in consequence of the present imperfect development of the quaternion calculus, the only known method of effecting this is to reduce the quaternion equation to a set of four ordinary differential equations of the first order. It may be interesting to form these equations.

Put q = w + ix +jy + kz, 7 ia +jb +kc, 330 then, by ordinary quaternion multiplication, we easily reduce the given equation to the following set : dt _ dw _ dx _ dy _ dz 2~~ W~X =~~ Y ~T where W = -x-y3$-z or X = . X= W + y(-z3$ Y=-X Y= iM + z-x$, Z= and = 1 [a (w* - x*-f- /) + 2x(ax + by + cz) + 2w (bz - cy)\ 3$ = n[b(w2 -x2 -yz - z2 } + 2y(ax + by+ cz) + 2w (ex - at)], = I [c (w 2 - a? -f- z2 ) + 2z(ax + by+ cz) + 2w (ay-bx~)]. W, X, Y, Z are thus homogeneous functions of w, x, y, z of the third degree.

Perhaps the simplest way of obtaining these equations is to translate the group of § 410 into w, x, y, z at once, instead of using the equation from which f and r\ are eliminated.

We thus see that

One obvious integral of these equations ought to be wz + x2 4- y 2 + z* = constant, which has been assumed all along. In fact, we see at once that identically, which leads to the above integral.

These equations appear to be worthy of attention, partly because of the homogeneity of the denominators W, X, Y, Z, but particularly as they afford (what does not appear to have been sought) the means of solving this celebrated problem at one step, that is, without the previous integration of Euler’s equations ( 2).

A set of equations identical with these, but not in a homogeneous form (being expressed, in fact, in terms of K, \, p, v of 375, instead of w, x, y, z), is given by Cayley (Camb. and Dub. 331 Math. Journal, vol. i. 1846), and completely integrated (in the sense of being reduced to quadratures) by assuming Euler’s equations to have been previously integrated. (Compare § 416.) Cayley’s method may be even more easily applied to the above equations than to his own ; and I therefore leave this part of the development to the reader, who will at once see (as in § 416) that , 23, ( correspond to DV o 2, o) 3 of the 77 type, § 412.

418. It may be well to notice, in connection with the formulae for direction cosines in § 375 above, that we may write = \[a (w*+ x2 -f - *2 ) + 26 (xy + wz) + 2c (xz - wy}\ 3$ = [2a (xy -wz) + b (w* -x* + f- z 2 ) + 2c (yz + wx)\ = i[2a (xz + wy) + 26 (yz -wx) + c (w* - x* -y* + z*)]. (j These expressions may be considerably simplified by the usual assumption, that one of the fixed unit-vectors (i suppose) is perpendicular to the invariable plane, which amounts to assigning definitely the initial position of one line in the body ; and which gives the relations 6-0, c = 0.

419. When forces act, 7 is variable, and the quantities a, 6, c will in general involve all the variables w, x, yy z, t, so that the equations of last section become much more complicated. The type, however, remains the same if 7 involves t only ; if it involve q we must differentiate the equation, put in the form 7 = and we thus easily obtain the differential equation of the second order ^ = 4F. (q- q) 3- + 2q_t_ ( V. q^j) g- ; if we recollect that, because q~ l q is a vector, we have Though the above formula is remarkably simple, it must, in the present state of the development of quaternions, be looked on as intractable, except in certain very particular cases.

332 420. Another mode of attacking the problem, at first sight entirely different from that in § 408, but in reality identical with it, is to seek the linear and vector function which expresses the Homogeneous Strain which the body must undergo to pass from its initial position to its position at time t. Let w = x, a being (as in § 408) the initial position of a vector of the body, OT its position at time t. In this case % is a linear and vector function. (See § 376.) Then, obviously, we have, ^l being the vector of some other point, which had initially the value at , $57^7 1 = S . (a particular case of which is and FOTOTI = V. These are necessary properties of the strain-function x depending on the fact that in the present application the system is rigid.

421. The kinernatical equation tzr = Veiff becomes %a = V. e^a (the function % being formed from % by the differentiation of its constituents with respect to t). Hamilton’s kinetic equation 2 . miff Vein = 7, becomes 2 . vti^p. V. e^a = 7. This may be written 2 . m (xaS . e^a - ea") = 7, or 2 . m (aS . a% e % -1 e . a2 ) = %~ J 7 where % is the conjugate of %. But, because S . xaXai = ^aai we have $, = S . % %! whatever be a and a1? so that x =x" Hence 2 . m (aS . a%~ J e X~IG a") = X J 7 or, by § 408 ^" ^X V

333 422. Thus we have, as the analogues of the equations in §§ 408, 409, and the former result %a = F. becomes %a = F. X^IX - = %F??a. This is our equation to determine %, 77 being supposed known. To find rj we may remark that ** = fc and ?=X~V But %%~IQC = a so that %x~ a + XX~ 1 a = - Hence t = ~ X~ l %X~ l V or (?) = These are the equations we obtained before. Having found from the last we have to find from the condition

423. We might, however, have eliminated 77 so as to obtain an equation containing ^ alone, and corresponding to that of § 410. For this purpose we have so that, finally, % -1 %a = ^ ~1 X~ 1 7a or ^a=^.%"W1% which may easily be formed from the preceding equation by putting -% IOL f r a and attending to the value of ^ given in last section.

424. We have given this process, though really a disguised form of that in §§ 408, 410, and though the final equations to which it leads are not quite so easily attacked in the way of integration as those there arrived at, mainly to shew how free a use we can make of symbolic functional operators in quaternions 334 without risk of error. It would be very interesting, however, to have the problem worked out afresh from this point of view by the help of the old analytical methods : as several new forms of long-known equations, and some useful transformations, would certainly be obtained.

425. As a verification, let us now try to pass from the final equation, in ^ alone, of § 423 to that of § 410 in q alone. We have, obviously, OT = qaq~ l = ^a, which gives the relation between q and ^. [It shews, for instance, that, as while S./3xoL = S. jSqaq 1 = S . aq~ l /3q, we have % /3 = q^ffq, and therefore that XX $ = # ( (fl @l) tf" 1 = or x = X~ l as above.] Differentiating, we have qaq 1 - qaq~ l qq~ l = tfa. Hence % -1 %a = f l ia "~ afl q Also c/r 1 x~ l y = F1 (fl so that the equation of § 423 becomes or, as a may have any value whatever, which, if we put Tq = constant as was originally assumed, may be written 2q = q^1 (ql yq), as in § 410.

C. Special Kinetic Problems.

426. To form the equation for Precession and Nutation. Let cr be the vector, from the centre of inertia of the earth, to a particle m of its mass: and let p be the vector of the disturbing body, whose mass is M. The vector-couple produced is evidently 335 mVcrp m no farther terms being necessary, since ^- is always small in the actual cases presented in nature. But, because a is measured from the centre of inertia, S . ma = 0. Also, as in § 408, // = S . m (aSap r 2 p). Thus the vector-couple required is L r Referred to coordinates moving with the body, $ becomes _j_ as in § 413, and § 413 gives Simplifying the value of J by assuming that the earth has two principal axes of equal moment of inertia, we have Be - (A - B) aSae = vector-constant + 3M (A - B) dt. This gives Sae = const. = H, whence e = Ha + ad, so that, finally, BVc-AOa = ^(A-B) VotpSap. The most striking peculiarity of this equation is that the form of the solution is entirely changed, not modified as in ordinary cases of disturbed motion, according to the nature of the value of p. Thus, when the right-hand side vanishes, we have an equation which, in the case of the earth, would represent the rolling of a cone fixed in the earth on one fixed in space, the angles of both being exceedingly small. 336 If p be finite, but constant, we have a case nearly the same as that of a top, the axis on the whole revolving conically about p. But if we assume the expression pr(j cos mt + k sin mi), (which represents a circular orbit described with uniform speed,) a revolves on the whole conically about the vector i, perpendicular to the plane in which p lies. (§§ 408—426, Trans. R 8. E., 18G8 9.)

427. To form the equation of motion of a simple pendulum, taking account of the earth’s rotation. Let a be the vector (from the earth’s centre) of the point of suspension, X its inclination to the plane of the equator, a the earth’s radius drawn to that point; and let the unit-vectors i,j,kbe fixed in space, so that i is parallel to the earth’s axis of rotation ; then, if co be the angular velocity of that rotation a = a [i sin X 4- (j cos a)t + k sin at) cos X] ......... (1). This gives a. = an (j sin cot + k cos cot) cos X = coVia .......................................... (2). Similarly a = CD Via = 2 (a ai sin X) .................. (3).

428. Let p be the vector of the bob m referred to the point of suspension, R the tension of the string, then if at be the direction of pure gravity m(a + p) = mg UOL^ R Up .................. (4), which may be written -Vw ............... (5). To this must be added, since r (the length of the string) is constant, Tp = r .............................. (6), and the equations of motion are complete.

429. These two equations (5) and (6) contain every possible case of the motion, from the most infinitesimal oscillations to the most rapid rotation about the point of suspension, so that it is necessary to adapt different processes for their solution in different cases. We take here only the ordinary Foucault case, to the degree of approximation usually given.

337 430. Here we neglect terms involving o 2 . Thus we write 8 = 0, and we write a for a t , as the difference depends upon the ellipticity of the earth. Also, attending to this, we have (7), where (by (6)) m = .............................. (8), and terms of the order TX* are neglected. With (7), (5) becomes so that, if we write - = nz .............................. (9), we have Fa(w + ?iV) = ..................... (10). Now, the two vectors ai a sin X and Via. have, as is easily seen, equal tensors ; the first is parallel to the line drawn horizontally northwards from the point of suspension, the second horizontally eastwards. Let, therefore, w = x (ai a sin X) + y Via ............ (11), which (x and y being very small) is consistent with (6). From this we have (employing (2) and (3), and omitting 2 ) CT = x (ai a sin X) + y Via. xw sin X Via. yco (a ai sin X), Hs x (ai OL sin X) + y Via. 2xa) sin X Via %yw (a ai sin X). With this (10) becomes Va \x (ai a sin X) + y Via 2xo) sin X Via 2?/w (a ai sin X) + n*x (ai a sin X) + tfyVia] = 0, or, if we note that V.aVia = a (ai a sin X), (as Zi/Q) sin X nz x] aVia + (y 2xco sin X +nz y)a(ai asinX) =0. This gives at once x + n2x + 2?/ sin X = 0] (12), y + tfy 2co% sin X = OJ which are the equations usually obtained; and of which the solution is as follows : If we transform to a set of axes revolving in the horizontal plane at the point of suspension, the direction of motion being from the T. Q. I. 22 338 positive (northward) axis of as to the positive (eastward) axis of y, with angular velocity fl, so that x = f cos fit r) sin fit] ._ _. y = (f sin 1U + ?? cos fltfj and omit the terms in H2 and in wO (a process justified by the results, see equation (15)), we have ( + ri*t;) cos fit - (77 + ft 2 ??) sin fit - 2y (1 - w sin X) = ...(14). sin +(*? + ^) cos HZ + 2#(H - G) sin X) = So that, if we put H = o sin X (15), we have simply |f + nz j; = (16) 1 + 11*11 = OJ the usual equations of elliptic motion about a centre of force in the centre of the ellipse. (Proc. R. 8. E., 1869.)

D. Geometrical and Physical Optics.

431. To construct a reflecting surface from which rays, emitted from a point, shall after reflection diverge uniformly, but horizontally. Using the ordinary property of a reflecting surface, we easily obtain the equation (j3 + oLVap\% P ~ P = By Hamilton’s grand Theory of Systems of Rays, we at once write down the second form Tp-T(/3 + a Vap) = constant. The connection between these is easily shewn thus. Let TX and r be any two vectors whose tensors are equal, then l (1 + SW1 ) (Chapter III. Ex. 2), whence, to a scalar factor pres, we have (^\2 T + VT T) T Hence, putting vr= U(fi + aVcLp) and T= ?7/?, we have from the first equation above 8. dp [Up +U(/3 + aVap)] = 0. 339 But d(/3 + a Vap) = a Vadp = -dp- aSadp, and S. so that we have finally which is the differential of the second equation above. A curious particular case is a parabolic cylinder, as may be easily seen geometrically. The general surface has a parabolic section in the plane of a, /3 ; and a hyperbolic section in the plane of ft, aft. It is easy to see that this is but a single case of a large class of integrable scalar functions, whose general type is $ . dp ( V P the equation of the reflecting surface ; while is the equation of the surface of the reflected wave : the integral of the former being, by the help of the latter, at once obtained in the form TpT(tr-p) = constant*.

432. We next take Fresnel’s Theory of Double Refraction, but merely for the purpose of shewing how quaternions simplify the processes required, and in no way to discuss the plausibility of the physical assumptions. Let far be the vector displacement of a portion of the ether, with the condition ^2 =-i a), the force of restitution, on Fresnel’s assumption, is t (cfiSi + VjSjiff + c Mtar) = tjysrt using the notation of Chapter V. Here the function / is the negative of that of Chapter IX. (the force of restitution and the displacement being on the whole towards opposite parts), and it is clearly self-conjugate, a2 , b*t c 2 are optical constants depending on the crystalline medium, and on the wave-length of the light, and may be considered as given. Fresnel’s second assumption is that the ether is incompressible, or that vibrations normal to a wave front are inadmissible. If, * Proc. E. S. E., 1870-71. 22-2 340 then, a be the unit normal to a plane wave in the crystal, we have of course 2 = -l .............................. (2), and SW = ............................... (3); but, and in addition, we have or $ . a-cr^-cr = ........................... (4). This equation (4) is the embodiment of Fresnel’s second assump tion, but it may evidently be read as meaning, the normal to the front, the direction of vibration, and that of the force of restitution are in one plane.

433. Equations (3) and (4), if satisfied by OT, are also satisfied by CTQC, so that the plane (3) intersects the cone (4) in two lines at right angles to each other. That is, for any given wave front there are two directions of vibration, and they are perpendicular to each other.

434. The square of the normal speed of propagation of a plane wave is proportional to the ratio of the resolved part of the force of restitution in the direction of vibration, to the amount of displacement, hence v2 = Stff^vf. Hence Fresnel’s Wave-surface is the envelop of the plane Sap = Svr_l_vr ........................ (5), with the conditions or 2 = 1 .............................. (1), 2 = -l .............................. (2), Sav = Q ................................. (3), S.aww = ....................... . ......... (4). Formidable as this problem appears, it is easy enough. From (3) and (4) we get at once, Hence, operating by S . r, X = Therefore ( + v*) = and ^f.a(0 + z; 5 )- 1 a = ..................... (C). 341 In passing, we may remark that this equation gives the normal speeds of the two rays whose fronts are perpendicular to a. In Cartesian coordinates it is the well-known equation 70 9 9 p +_?!!_ +^!_= o. By this elimination of -cr, our equations are reduced to = ........................ (6), v = -Sp ........................... (5), 2 = -l .............................. (2). They give at once, by § 326, ( + V^OL + vpSa (0 + v*)- 2a = ha. Operating by S . a we have Substituting for h, and remarking that Sa(l + v 2ra = - because (/ is self-conjugate, we have This gives at once, by rearrangement, v (t + vT1* = (#- pTp- Hence - = Operating by S .p on this equation we have SpQ-ffp l ..................... (7), which is the required equation. [It will be a good exercise for the student to translate the last ten formulae into Cartesian coordinates. He will thus reproduce almost exactly the steps by which Archibald Smith* first arrived at a simple and symmetrical mode of effecting the elimination. Yet, as we shall presently see, the above process is far from being the shortest and easiest to which quaternions conduct us.]

435. The Cartesian form of the equation (7) is not the usual one. It is, of course, * Cambridge Phil. Trans., 1835. 342 But write (7) in the form and we have the usual expression oV ^ by cV_ The last-written quaternion equation can also be put into either of the new forms or

436. By applying the results of §§ 183, 184 we may introduce a multitude of new forms. We must confine ourselves to the most simple ; but the student may easily investigate others by a process precisely similar to that which follows. Writing the equation of the wave as where we have g = p~*, we see that it may be changed to sp (^1 + hrl p = o, if mSpjp = glip* = h. Thus the new form is Sp((F l -mSplprp = ..................... (1). Here m = -,--,, Sp^p = aV + Vtf + cV, Ct C and the equation of the wave in Cartesian coordinates is, putting a? y* z* ^ + - * + ** - c § 437. By means of equation (1) of last section we may easily prove Pliicker’s Theorem : The Wave-Surface is its own reciprocal with respect to the ellipsoid whose equation is 343 The equation of the plane of contact of tangents to this surface from the point whose vector is p is The reciprocal of this plane, with respect to the unit-sphere about the origin, has therefore a vector cr where Hence p = p. ~ and when this is substituted in the equation of the wave we have for the reciprocal (with respect to the unit-sphere) of the reciprocal of the wave with respect to the above ellipsoid, 1 \ -1 O(J(p (T 1 (T == 0. in / This differs from the equation (1) of last section solely in having 0" 1 instead of (/, and (consistently with this) l/m instead of m. Hence it represents the index-surface. The required reciprocal of the wave with reference to the ellipsoid is therefore the wave itself.

438. Hamilton has given a remarkably simple investigation of the form of the equation of the wave-surface, in his Elements, p. 736, which the reader may consult with advantage. The following is essentially the same, but several steps of the process, which a skilled analyst would not require to write down, are retained for the benefit of the learner. Let % = -! (1) be the equation of any tangent plane to the wave, i.e. of any wavefront. Then //, is the vector of wave-slowness, and the normal velocity of propagation is therefore 1/Tfj,. Hence, if TZ be the vector direction of displacement, /J,~*TZ- is the effective component of the force of restitution. Hence, (/OT denoting the whole force of restitution, we have and, as CT is in the plane of the wave-front, = 0, "2 )~ lp = (2). 344 This is, in reality, equation (6) of § 434. It appears here, however, as the equation of the Index-Surface, the polar reciprocal of the wave with respect to a unit-sphere about the origin. Of course the optical part of the problem is now solved, all that remains being the geometrical process of § 328.

439. Equation (2) of last section may be at once transformed, by the process of § 435, into Let us employ an auxiliary vector T=G*2 -.-, whence ^ = (/** - (/T^r ........................ (1). The equation now becomes SHT = ! .............................. (2), or,by(l), ^-Sr^r = l ........................ (3). Differentiating (3), subtract its half from the result obtained by operating with S . r on the differential of (1). The remainder is But we have also (§ 328) Spdfi = 0, and therefore, since dp has an infinite number of values, Xp = yL6T 2 T, where x is a scalar. This equation, with (2), shews that Srp = () .............................. (4). Hence, operating on it by 8 . p, we have by (1) of last section V= -T* and therefore p 1 = //, + r"1 . This gives p~ 2 = ^ - T "2 . Substituting from these equations in (1) above, it becomes T- -^^+T-2 --, or T-($-*-p*)-V. Finally, we have for the required equation, by (4), Bpyr-pTpi-o, or, by a transformation already employed,

345 440. It may assist the student in the practice of quaternion analysis, which is our main object, if we give a few of these investigations by a somewhat varied process. Thus, in § 432, let us write as in § 180, cfiSi-v + tfjSjw + tfkSk-v = \ Sp!^ + p S^v? pr ix. We have, by the same processes as in § 432, 8 . uroWSp v + S . isroLfi SXfw = 0. This may be written, 50 far as the generating lines we require are concerned, S.vaV. \ V/JL = = 8. since ora is a vector. Or we may write S.fi V. OTX -Bra = = 8. Equations (1) denote two cones of the second order which pass through the intersections of (3) and (4) of § 432. Hence their intersections are the directions of vibration.

441. By (1) we have .(1). Hence rXV, a, fjf are coplanar ; and, as w is perpendicular to a, it is equally inclined to FX a and F//a. For, if L, M, A be the projections of X , ///, a on the unit sphere, BG the great circle whose pole is A, we are to find for the ^ projections of the values of OT on the sphere points P and P , such that if LP be produced till Q may lie on the great circle AM. Hence, evidently, and which proves the proposition, since the projections of FX a and F//a on the sphere are points b and c in BC, distant by quadrants from G and B respectively.

346 442. Or thus, Si*a. = 0, S . vrV . aX vT/jf = 0, therefore XTZ = V. aF. = - F. XV// - Hence (x - a?) w = (V + aaX ) / + (/ Operate by $ . X , and we have (x + SXaSfi a) SxV = [XV - SVa Hence by symmetry, J _~ and as S-OL = 0, /a UVpfa).

443. The optical interpretation of the common result of the last two sections is that the planes of polarization of the two rays whose wave-fronts are parallel, bisect the angles contained by planes passing through the normal to the wave-front and the vectors (optic axes) X , fju.

444. As in § 434, the normal speed is given by [This transformation, effected by means of the value of r in § 442, is left to the reader.] Hence, if viy v 2 be the velocities of the two waves whose normal is a, oc sin X a sin /u/a. That is, the difference of the squares of the speeds of the two waves varies as the product of the sines of the angles between the normal to the wave-front and the optic axes (X , //).

347 445. We have, obviously, (T2 - S2 ) . V\ OL Vp a =TV. V\ OL VH OL = S2 . \ Hence v* =p + (T S) . VXaVpa. The equation of the index surface, for which is therefore 1 = - p p* + (T S) . VX p Vp p. This will, of course, become the equation of the reciprocal of the index-surface, i.e. the wave-surface, if we put for the function _/_ its reciprocal : i.e. if in the values of X , //, , p we put I/a, 1/6, 1/c for a, 6, c respectively. We have then, and indeed it might have been deduced even more simply as a transformation of § 434 (7), as another form of the equation of Fresnel’s wave. If we employ the i, K transformation of § 128, this may be written, as the student may easily prove, in the form (V - ij = S*(i- K)p + (TViP

446. We may now, in furtherance of our object, which is to give varied examples of quaternions, not complete treatment of any one subject, proceed to deduce some of the properties of the wavesurface from the different forms of its equation which we have given.

447. Fresnel’s construction of the wave by points. From § 290 (4) we see at once that the lengths of the principal semidiameters of the central section of the ellipsoid Spj- p = 1, by the plane Sap = 0, are determined by the equation 3.a(l- l -p-*)- 1a = 0. If these lengths be laid off along a, the central perpendicular to the cutting plane, their extremities lie on a surface for which a = Up, and Tp has values determined by the equation. Hence the equation of the locus is as in §§ 434, 439. 348 Of course the index-surface is derived from the reciprocal ellipsoid SpQp = 1 by the same construction.

448. Again, in the equation l=-pp*+(TS).V\P Vfj,p, suppose V\p = 0, or Vjj,p = 0, we obviously have ,U\ tfyu, P = -7- r P = , , \/p *Jp and there are therefore four singular points. To find the nature of the surface near these points put U\ P = -/- + , Vp where TVT is very small, and reject terms above the first order in TOT. The equation of the wave becomes, in the neighbourhood of the singular point, 2pS\v + S.V. \V\fji = T. V\iz V\p, which belongs to a cone of the second order.

449. From the similarity of its equation to that of the wave, it is obvious that the index-surface also has four conical cusps. As an infinite number of tangent planes can be drawn at such a point, the reciprocal surface must be capable of being touched by a plane at an infinite number of points ; so that the wave-surface has four tangent planes which touch it along ridges. To find their form, let us employ the last form of equation of the wave in § 445. If we put TKP = TVKp (1), we have the equation of a cone of the second degree. It meets the wave at its intersections with the planes fl((.-*)/ = ("- ) (2). Now the wave-surface is touched by these planes, because we cannot have the quantity on the first side of this equation greater in absolute magnitude than that on the second, so long as p satisfies the equation of the wave. 349 That the curves of contact are circles appears at once from (1) and (2), for they give in combination )p........................ (3), the equations of two spheres on which the curves in question are situated. The diameter of this circular ridge is = (a* - V) (b* - f). [Simple as these processes are, the student will find on trial that the equation V = 0, gives the results quite as simply. For we have only to examine the cases in which p~ z has the value of one of the roots of the symbolical cubic in /~\ In the present case Tp = b is the only one which requires to be studied.]

450. By § 438, we see that the auxiliary vector of the succeeding section, viz. T =(/* -- ) = (- -p-)V, is parallel to the direction of the force of restitution, ^t7. Hence, as Hamilton has shewn, the equation of the wave, in the form (4) of § 439, indicates that the direction of the force of restitution is perpendicular to the ray. Again, as for any one versor of a vector of the wave there are two values of the tensor, which are found from the equation we see by § 447 that the lines of vibration for a given plane front are parallel to the axes of any section of the ellipsoid made by a plane parallel to the front ; or to the tangents to the lines of curvature at a point where the tangent plane is parallel to the wave-front.

451. Again, a curve which is drawn on the wave-surface so as to touch at each point the corresponding line of vibration has 350 Hence Sfypdp = 0, or Sp(fp = C, so that such curves are the intersections of the wave with a series of ellipsoids concentric with it.

452. For curves cutting at right angles the lines of vibration we have Hence Spdp = 0, or Tp = C, so that the curves in question lie on concentric spheres. They are also spherical conies, because where Tp = C the equation of the wave becomes s. P (4- + ryv = o, the equation of a cyclic cone, whose vertex is at the common centre of the sphere and the wave-surface, and which cuts them in their curve of intersection. (§ 432—452, Quarterly Math. Journal, 1859.) The student may profitably compare, with the preceding investi gations, the (generally) very different processes which Hamilton (in his Elements) applies to this problem.

E. Electrodynamics.

453. As another example we take the case of the action of electric currents on one another or on magnets; and the mutual action of permanent magnets.

A comparison between the processes we employ and those of Ampère (Théorie des Phénomènes Electrodynamiques) will at once shew how much is gained in simplicity and directness by the use of quaternions.

The same gain in simplicity will be noticed in the investigations of the mutual effects of permanent magnets, where the resultant forces and couples are at once introduced in their most natural and direct forms.

454. Ampère’s experimental laws may be stated as follows :

I. Equal and opposite currents in the same conductor produce equal and opposite effects on other conductors.

351 II. The effect of a conductor bent or twisted in any manner is equivalent to that of a straight one, provided that the two are traversed by equal currents, and the former nearly coincides with the latter.

III. No closed circuit can set in motion an element of a circular conductor about an axis through the centre of the circle and perpendicular to its plane.

IV. In similar systems traversed by equal currents the forces are equal.

To these we add the assumption that the action between two elements of currents is in the straight line joining them. [In a later section (§ 473) other assumptions will be made in place of this.] We also take for granted that the effect of any element of a current on another is directly as the product of the strengths of the currents, and of the lengths of the elements.

455. Let there be two closed currents whose strengths are a and al ; let a, ax be elements of these, a being the vector joining their middle points. Then the effect of a on a l must, when resolved along alf be a complete differential with respect to a (i.e. with respect to the three independent variables involved in a), since the total resolved effect of the closed circuit of which a is an element is zero by III.

Also by I, II, the effect is a function of Ta, Sa.a, Saa.^ and SCL CL^ since these are sufficient to resolve a and al into elements parallel and perpendicular to each other and to a. Hence the mutual effect is a^Uaf^a, Saa , SOL\), and the resolved effect parallel to al is aa^SUa^Uaf. Also, that action and reaction may be equal in absolute magnitude, / must be symmetrical in 8a.a and $aar Again, a (as differential of a) can enter only to the first power, and must appear in each term off.

Hence /= A8a\ + BScm Sou^. But, by IV, this must be independent of the dimensions of the system. Hence A is of 2 and B of 4 dimensions in To.. Therefore - [ASaa.Safa, + BSaa aaJ 352 is a complete differential, with respect to a, if da. = of. Let G ~Tc? where G is a constant depending on the units employed, therefore G B and the resolved effect Cad -. S O.O.. = To? = * TaTc? ^ + The factor in brackets is evidently proportional in the ordinary notation to sin 6 sin cos co | cos 6 cos 6 . *

456. Thus the whole force is Caa^a j $2aat _ Caa^a. , 8*0.0. ~ l ~ 3 as we should expect, d^a. being = ar [This may easily be transformed into which is the quaternion expression for Ampère’s well-known form.]

457. The whole effect on at of the closed circuit, of which a is an element, is therefore Oaa rv a J ZV) between proper limits. As the integrated part is the same at both limits, the effect is and depends on the form of the closed circuit.

353 458. This vector 0, which is of great importance in the whole theory of the effects of closed or indefinitely extended circuits, cor responds to the line which is called by Ampère “directrice de l’action électrodynamique.” It has a definite value at each point of space, independent of the existence of any other current.

Consider the circuit a polygon whose sides are indefinitely small ; join its angular points with any assumed point, erect at the latter, perpendicular to the plane of each elementary triangle so formed, a vector whose length is a/r, where is the vertical angle of the triangle and r the length of one of the containing sides ; the sum of such vectors is the “directrice” at the assumed point.

[We may anticipate here so far as to give another expression for this important vector in terms of processes to be explained later.

We have, by the formula (for a closed curve) of § 497 below, (where ds is an element of any surface bounded by the circuit, Uv its unit normal)

But the last integral is obviously the whole spherical angle, II suppose, subtended by the circuit at the origin, and (unless Tp = 0) we have (§ 145) Hence, generally, /e=-vfl. Thus II may be considered as representing a potential, for which ft is the corresponding force.

This is a “many-valued” function, altering by 4-Tr whenever we pass through a surface closing the circuit. For if o- be the vector of a closed curve, the work done against /3 during the circuit is fSjBdo- = - J SdoVQ, = fdtt.

The last term is zero if the curve is not linked with the circuit, but increases by + 4-n- for each linkage with the circuit.] T. Q. I. 23

354 459. The mere form of the result of § 457 shews at once that if the element ax be turned about its middle point, the direction of the resultant action is confined to the plane whose normal is {3. Suppose that the element at is forced to remain perpendicular to some given vector 8, we have Saf = 0, and the whole action in its plane of motion is proportional to TV.BVctfi. But V . SVafi = - afilSS. Hence the action is evidently constant for all possible positions of otj ; or The effect of any system of closed currents on an element of a conductor which is restricted to a given plane is (in that plane) independent of the direction of the element.

460. Let the closed current be plane and very small. Let e (where Te = 1) be its normal, and let 7 be the vector of any point within it (as the centre of inertia of its area) ; the middle point of 4 being the origin of vectors. Let a. = 7 + p ; therefore a = p, to a sufficient approximation. Now (between limits) jVpp = where A is the area of the closed circuit. Also generally fVyp Syp = J (%F7p + y 7. = (between limits) A 7 Vye. Hence for this case

355 461. If, instead of one small plane closed current, there be a series of such, of equal area, disposed regularly in a tubular form, let x be the distance between two consecutive currents measured along the axis of the tube ; then, putting y = xe, we have for the whole effect of such a set of currents on a CAaa f/y 2x 1 A?V + Ty5 CAaa. Va..y ,, ,. . ^ -^ V (between proper limits). If the axis of the tubular arrangement be a closed curve this will evidently vanish. Hence a closed solenoid exerts no influence on an element of a conductor. The same is evidently true if the solenoid be indefinite in both directions. It the axis extend to infinity in one direction, and yn be the vector of the other extremity, the effect is and is therefore perpendicular to the element and to the line joining it with the extremity of the solenoid. It is evidently inversely as Ty* and directly as the sine of the angle contained between the direction of the element and that of the line joining the latter with the extremity of the solenoid. It is also inversely as x, and therefore directly as the number of currents in a unit of the axis of the solenoid.

462. To find the effect of the whole circuit whose element is otj on the extremity of the solenoid, we must change the sign of the above and put al = y , therefore the effect is CAaal f V% % 2x ] Ty* an integral of the species considered in § 458, whose value is easily assigned in particular cases.

463. Suppose the conductor to be straight) and indefinitely extended in both directions. Let hO be the vector perpendicular to it from the extremity of the solenoid, and let the conductor be || 77, where T6 = Trj = 1. Therefore y = h + yrj (where y is a scalar), 23-2 356 and the integral in § 462 is -Jr* The whole effect is therefore _CAaaJ xh and is thus perpendicular to the plane passing through the conductor and the extremity of the solenoid, and varies inversely as the distance of the latter from the conductor. This is exactly the observed effect of an indefinite straight current on a magnetic pole, or particle of free magnetism.

464. Suppose the conductor to be circular, and the pole nearly in its axis. [This is not a proper subject for Quaternions.] Let EPD be the conductor, AB its axis, and G the pole ; BG perpendicular to AB, and small in comparison with AE = h the radius of the circle. Let AB be aj, BC=bk, AP = h(jx + ley) where \ = \ . \EAP=\ . \ B. y] lsmj lsmj Then CP = y = a^i -\-bk-h (jx + ky). And the effect on G x l-fJ-j- t where the integral extends to the whole circuit.

357 465. Suppose in particular C to be one pole of a small magnet or solenoid CO whose length is 21, and whose middle point is at G and distant a from the centre of the conductor. Let /.CGB = A. Then evidently ax = a 4- 1 cos A, b = I sin A. Also the effect on C becomes, if a/ 2 -f b 2 + h2 = A2 , h1 f/r [ij J 3 I ^ IV * . //" -" 1 ~/ I w,.i/it/ i t-"-I since for the whole circuit If we suppose the centre of the magnet fixed, the vector axis of the couple produced by the action of the current on C is IV. (i cos A 4- sin A) I -^^ sin A . L 362 . 15W 3a,6 cos A A3 A* 2 If A} &c. be now developed in powers of , this at once becomes vh*l sin A . f _ 6aZ cos A 15aT cos2 A / I 9 . T (a 2 -f ^) "I a2 + W (a 2 + A2 ) 2 a2 + /i 2 _3^sin 2 A 15 /iT sin2 A (a + Zcos A)^cos A ~ 5a^cosA\) aM- A2 h (a 2 + A2 ) 2 ~ a* + h* ( ~?"+l ~ / j Putting ^ for Z and changing the sign of the whole to get that for pole C , we have for the vector axis of the complete couple A . which is almost exactly proportional to sin A, if 2a be nearly equal to h and I be small. (See Ex. 15 at end of Chapter.) On this depends a modification of the tangent galvanometer. (Bravais, Ann. de Ghimie, xxxviii. § 309.)

358 466. As before, the effect of an indefinite solenoid on at is CAaa1 VOL*/ ~^T~T^ Now suppose at to be an element of a small plane circuit, 8 the vector of the centre of inertia of its area, the pole of the solenoid being origin. Let 7 = 8 -f p, then ax = p. The whole effect is therefore CAaa1 GAAaa where Al and e t are, for the new circuit, what A and e were for the former. Let the new circuit also belong to an indefinite solenoid, and let S be the vector joining the poles of the two solenoids. Then the mutual effect is GAA aa _CAAlaal S UB ~ which is exactly the mutual effect of two magnetic poles. Two finite solenoids, therefore, act on each other exactly as two magnets, and the pole of an indefinite solenoid acts as a particle of free magnetism.

467. The mutual attraction of two indefinitely small plane closed circuits, whose normals are e and e lt may evidently be deduced by twice differentiating the expression U8/T for the mutual action of the poles of two indefinite solenoids, making dS in one differentiation [| e and in the other || e lt But it may also be calculated directly by a process which will give us in addition the couple impressed on one of the circuits by the other, supposing for simplicity the first to be circular. [In the sketch we are supposed to be looking at the faces turned towards one another.] 359 Let A and B be the centres of inertia of the areas of A and B, e and e, vectors normal to their planes, cr any vector radius of B =0. Then whole effect on cr , by §§ 457, 460, -A VA * V7e - But between proper limits, for generally fV(r rjS0a =-%( VrjaSOo- +V.rjV. 6/Vaa). Hence, after a reduction or two, we find that the whole force exerted by A on the centre of inertia of the area of B reh This, as already observed, may be at once found by twice TTO differentiating 7*7 - In the same way the vector moment, due to A, about the centre of inertia of B, C - 777 These expressions for the whole force of one small magnet on the centre of inertia of another, and the couple about the latter, seem to be the simplest that can be given. It is easy to deduce 360 from them the ordinary forms. For instance, the whole resultant couple on the second magnet may easily be shewn to coincide with that given by Ellis (Gamb. Math. Journal, iv. 95), though it seems to lose in simplicity and capability of interpretation by such modifications.

468. The above formulae shew that the whole force exerted by one small magnet M, on the centre of inertia of another m consists of four terms which are, in order, 1st. In the line joining the magnets, and proportional to the cosine of their mutual inclination. 2nd. In the same line, and proportional to five times the product of the cosines of their respective inclinations to this line. 3rd and 4th. Parallel to ^ and proportional to the cosine of the inclination of \ ! to the joining line. All these forces are, in addition, inversely as the fourth power of the distance between the magnets. For the couples about the centre of inertia of m we have 1st. A couple whose axis is perpendicular to each magnet, and which is as the sine of their mutual inclination. 2nd A couple whose axis is perpendicular to m and to the line joining the magnets, and whose moment is as three times the product of the sine of the inclination of m, and the cosine of the inclination of M, to the joining line. In addition these couples vary inversely as the third power of the distance between the magnets. [These results afford a good example of what has been called the internal nature of the methods of quaternions, reducing, as they do at once, the forces and couples to others independent of any lines of reference, other than those necessarily belonging to the system under consideration. To shew their ready applicability, let us take a Theorem due to Gauss.]

361 469. If two small magnets be at right angles to each other, the moment of rotation of the first is approximately twice as great when the axis of the second passes through the centre of the first, as when the axis of the first passes through the centre of the second. In the first case e II $ _!_ e t ; C 2C" therefore moment = T (eel - 3e^) =^^eer In the second e x || @ J_ e ; G therefore moment = ^- Tee Hence the theorem.

470. Again, we may easily reproduce the results of § 467, if for the two small circuits we suppose two small magnets perpen dicular to their planes to be substituted. /3 is then the vector joining the middle points of these magnets, and by changing the tensors we may take 2e and 2e x as the vector lengths of the magnets. Hence evidently the mutual effect oc which is easily reducible to as before, if smaller terms be omitted. If we operate with F. e x on the two first terms of the unreduced expression, and take the difference between this result and the same with the sign of e : changed, we have the whole vector axis of the couple on the magnet 2ev which is therefore, as before, seen to be proportional to

471. Let F (7) be the potential of any system upon a unit particle at the extremity of 7. Then we have Svdy = 0, where v is a vector normal giving the force in direction and magnitude (§ 148). 362 Now by § 460 we have for the vector force exerted by a small plane closed circuit on a particle of free magnetism the expression A / SySye\ T*) merging in A the factors depending on the strength of the current and the strength of magnetism of the particle. Hence the potential is ASey Zy ^ area of circuit projected perpendicular to 7 7p 2 Trf x spherical opening subtended by circuit. The constant is omitted in the integration, as the potential must evidently vanish for infinite values of Ty. By means of Ampère’s idea of breaking up a finite circuit into an indefinite number of indefinitely small ones, it is evident that the above result may be at once ex tended to the case of such a finite closed circuit.

472. Quaternions give a simple method of deducing the well-known property of the Magnetic Curves. Let A, A be two equal magnetic poles, whose vector distance, 2a, is bi- A~~ o ~A f sected in 0, QQ an indefinitely small magnet whose length is 2/o , where p = OP. Then evidently, taking moments, a)j , V(p-*)p * where the upper or lower sign is to be taken according as the poles are like or unlike. [This equation may also be obtained at once by differentiating the equation of the equipotential surface, + T, X = COnSt" T (p + a) T (p - a and taking p parallel to its normal (§ 148).] 363 Operate by 8 . Vap, Sap (p + a) 2 So. (p + a) So (p + a) .., ~.n/ -T3 - = same with - a}, -L (p + a) or S.aFf- J CT(/o + a)= {same with -a}, i.e. SadU(p + a) = + $acZ U(p - a), a {#"(/ + a) + tT(p - a)} = const., or cos /. OAP cos OA P = const, the property referred to. If the poles be unequal, one of the terms to the left must be multiplied by the ratio of their strengths. (§§ 453—472, Quarterly Math. Journal, 1860.)

F. General Expressions for the Action between Linear Elements.

473. The following general investigation of different possible expressions for the mutual action between elements of linear conductors is taken from Proc. R. S. E. 1873 4. Ampère’s data for closed currents are briefly given in § 454 above, and are here referred to as I, II, III, IV, respectively. (a) First, let us investigate the expression for t\\e force exerted by one element on another. Let a be the vector joining the elements ap a , of two circuits ; then, by I, II, the action of a. v on a is linear in each of at , a , and may, therefore, be expressed as K where / is a linear and vector function, into each of whose con stituents otj enters linearly. The resolved part of this along a is 8. Z7a K and, by III, this must be a complete differential as regards the circuit of which a, is an element. Hence, _/_a = -(8. a,V) ^a + Fa x P where \|r and ^ are linear and vector functions whose constituents involve a only. That this is the case follows from the fact that (fa is homogeneous and linear in each of al5 a . It farther 364 follows, from IV, that the part of /a which does not disappear after integration round each of the closed circuits is of no dimen sions in To., To. , Ta^ Hence % is of 2 dimensions in To., and thus i - ~2V " 5V ~W~ where jo, q, r are numbers. Hence we have ofaSaoL, qV*^ rV.a Vaa, Change the sign of a in this, and interchange of and a,, and we get the action of a! on ar This, with a and a t again interchanged, and the sign of the whole changed, should reproduce the original expression since the effect depends on the relative, not the abso lute, positions of a, a.v a . This gives at once, p = 0, q = 0, and r V. a Fact with the condition that the first term changes its sign with a, and thus that i|ra = aSaa F (To) + a! which, by change of F, may be written where /and F are any scalar functions whatever. Hence ja = -S (cqV) [aS (a V)f(Ta) + a F (To)] +~^-1 which is the general expression required. (6) The simplest possible form for the action of one currentelement on another is, therefore, 2V Here it is to be observed that Ampère’s directrice for the circuit a. is /Too, "JTW the integral extending round the circuit ; so that, finally, 365 (c) We may obtain from the general expression above the absolutely symmetrical form, if we assume /(2k) = const, Here the action of a on a t is parallel and equal to that of at on a. The forces, in fact, form a couple, for a is to be taken negatively for the second and their common direction is the vector drawn to the corner a of a spherical triangle abc, whose sides ab, be, ca in order are bisected by the extremities of the vectors Ua, Ua, Ua^ Compare Hamilton’s Lectures on Quaternions, §§ 223—227. (d) To obtain Ampère’s form for the effect of one element on another write, in the general formula above, and we have !. - a.* [_s*r[^r.*vmi To V. a Faa, * / 20 3 o o "\ g I Ct oCtjtt Q ottft oOCOCj 1 , ToI \ -^ / 5 a_J^/$.F^jrvfj which are the usual forms. (e) The remainder of the expression, containing the arbitrary terms, is of course still of the form - S (a,V, [aS (a V)/(Ta) + a F (2V.)]. In the ordinary notation this expresses a force whose com ponents are proportional to (Note that, in this expression, r is the distance between the elements.) (2) Parallel to a ^. 366 (3) Parallel to - ^ . as If we assume /= F= Q, we obtain the result given by Clerk- Maxwell (Electricity and Magnetism, § 525), which differs from the above only because he assumes that the force exerted by one element on another, when the first is parallel and the second perpendicular to the line joining them, is equal to that exerted when the first is perpendicular and the second parallel to that line. (/) What precedes is, of course, only a particular case of the following interesting problem : Required the most general expression for the mutual action of two rectilinear elements, each of which has dipolar symmetry in the direction of its length, and which may be resolved and compounded according to the usual kinematical law. The data involved in this statement are equivalent to I and II of Ampère’s data above quoted. Hence, keeping the same notation as in (a) above, the force exerted by ctj on a must be expressible as (j)OL where (f is a linear and vector function, whose constituents are linear and homogeneous in at ; and, besides, involve only a. By interchanging a t and a , and changing the sign of a, we get the force exerted by a on ar If in this we again interchange at and a, and change the sign of the whole, we must obviously repro duce / . Hence we must have $ changing its sign with a, or (/ = PaSa/ + QaSaaficta. + Rafiaa! + RaSctctv where P, Q, R, R are functions of Ta only. (g) The vector couple exerted by a t on a must obviously be expressible in the form V, a waj, where or is a new linear and vector function depending on a alone. Hence its most general form is where P and Q are functions of To. only. The form of these func tions, whether in the expression for the force or for the couple, depends on the special data for each particular case. Symmetry shews that there is no term such as 367 (ti) As an example, let al and a be elements of solenoids or of uniformly and linearly magnetised wires, it is obvious that, as a closed solenoid or ring-magnet exerts no external action, Thus we have introduced a different datum in place of Ampère’s No. III. But in the case of solenoids the Third Law of Newton holds hence where % is a linear and vector function, and can therefore be of no other form than Now two solenoids, each extended to infinity in one direction, act on one another like two magnetic poles, so that (this being our equivalent for Ampère’s datum No. TV.) Hence the vector force exerted by one small magnet on another is . (i) For the couple exerted by one element of a solenoid, or of a uniformly and longitudinally magnetised wire, on another, we have of course the expression V. * ** where * is some linear and vector function. Here, in the first place, it is obvious that for the couple vanishes for a closed circuit of which a t is an element, and the integral of wo^ must be a linear and vector function of a alone. It is easy to see that in this case F(Ta) oc (Taf. (j) If, again, at be an element of a solenoid, and a an element of current, the force is = _ Sa^V . Tfra t where ^a = Pa + QaSacL + RVoaf. 368 But no portion of a solenoid can produce a force on an element of current in the direction of the element, so that whence P = 0, Q = 0, and we have = SatV (RVaa. ). This must be of 1 linear dimensions when we integrate for the effect of one pole of a solenoid, so that 7? P ** = 7/r~3 If the current be straight and infinite each way, its equation being where Ty = l and Sfty = 0, we have, for the whole force exerted on it by the pole of a solenoid, the expression /+ 00 dx Pftv I ^= ~ 2PP 7i which agrees with known facts. () Similarly, for the couple produced by an element of a solenoid on an element of a current we have where and it is easily seen that ra. (I) In the case first treated, the couple exerted by one currentelement on another is, by (g), V. CL K^, where, of course, ^^ are the vector forces applied at either end of a . Hence the work done when a changes its direction is with the condition S . a So! = 0. So far, therefore, as change of direction of a alone is concerned, the mutual potential energy of the two elements is of the form 8 . 369 This gives, by the expression for OT in (g), the following value Hence, integrating round the circuit of which cq is an element, we have (§ 495 below) f(PSa\ + QSaa SaaJ =ffdsfi. Up,V (Pa! + QctSaa), P where =_ + Q. Integrating this round the other circuit we have for the mutual potential energy of the two, so far as it depends on the expression above, the value ffdsfi. UpJVota. 3 = -ffdSlS. UpJJds V. UP (24 + TOL) + Sa Uv Sa UP. But, by Ampère’s result, that two closed circuits act on one another as two magnetic shells, it should be ffds, ffds S . Uv.VS . Uv V ~ = ffds, ffds S . Uv, UP + 3^a UP SOL UP, Comparing, we have gving J) = - * = which are consistent with one another, and which lead to Hence, if we put 1 n weget p T. Q. I. 24 370 and the mutual potential of two elements is of the form , Sao, , . Sao. Seta. which is the expression employed by Helmholtz in his paper Ueber die Bewegungsgleichungen der Electricitdt, Crelle, 1870, p. 76.

G. Application of V to certain Physical Analogies.

474. The chief elementary results into which V enters, in connection with displacements, are given in § 384 above. The following are direct applications. Thus, if o- be the vector-displacement of that point of a homogeneous elastic solid whose vector is p, we have, p being the consequent pressure produced, whence SpV 2a = SpVp = $p, a complete differential (2). Also, generally, p = cSVa, and if the solid be incompressible SVo- = Q (3). Thomson has shewn (Camb. and Dub. Math. Journal, ii. p. 62), that the forces produced by given distributions of matter, electricity, magnetism, or galvanic currents, can be represented at every point by displacements of such a solid producible by external forces. It may be useful to give his analysis, with some additions, in a quaternion form, to shew the insight gained by the simplicity of the present method.

475. Thus, if Sa-Sp = 5 , we may write each equal to This gives the vector-force exerted by one particle of matter or free electricity on another. This value of cr evidently satisfies (2) and (3). Again, if S . SpVa = 8 ^ , either is equal to 371 Here a particular case is Vao ~ Tf which is the vector-force exerted by an element a of a current upon a particle of magnetism at p. (§ 461.)

476. Also, by § 146 (3), and we see by §§ 460, 461 that this is the vector-force exerted by a small plane current at the origin (its plane being perpendicular to a) upon a magnetic particle, or pole of a solenoid, at p. This expression, being a pure vector, denotes an elementary rotation caused by the distortion of the solid, and it is evident that the above value of cr satisfies the equations (2), (3), and the distortion is therefore producible by external forces. Thus the effect of an element of a current on a magnetic particle is expressed directly by the displacement, while that of a small closed current or magnet is represented by the vector-axis of the rotation caused by the displacement.

477. Again, let SSPV^ = S^- It is evident that a satisfies (2), and that the right-hand side of the above equation may be written Hence a particular case is Tp* and this satisfies (3) also. Hence the corresponding displacement is producible by external forces, and Vr is the rotation axis of the element at p, and is seen as before to represent the vector-force exerted on a particle of magnetism at p by an element a of a current at the origin.

478. It is interesting to observe that a particular value of cr in this case is as may easily be proved by substitution. 24-2 372 Again, if S8pr - j~~ , = we have evidently cr = V _- . Now, as ^~ is the potential of a small magnet a, at the origin, on a particle of free magnetism at p, cr is the resultant magnetic force, and represents also a possible distortion of the elastic solid by external forces, since Vcr =W= 0, and thus (2) and (3) are both satisfied. (Proc. R S, E. 1862.)

H. Elementary Properties of V.

479. In the next succeeding sections we commence with a form of definition of the operator V somewhat different from that of Hamilton (§ 145), as we shall thus entirely avoid the use of Cartesian coordinates. For this purpose we write where a is any unit-vector, the meaning of the right-hand operator (neglecting its sign) being the rate of change of the function to which it is applied per unit of length in the direction of the vector a. If a be not a unit-vector we may treat it as a vectorvelocity, and then the right-hand operator means the rate of change per unit of time due to the change of position. Let a, /3, 7 be any rectangular system of unit-vectors, then by a fundamental quaternion transformation V - - aaSaaV - /3/SSV - 7^7V = aada + ft + y y , which is identical with Hamilton’s form so often given above. (Lectures, § 620.)

480. This mode of viewing the subject enables us to see at once that V is an Invariant, and that the effect of applying it to any scalar function of the position of a point is to give its vector of most rapid increase. Hence, when it is applied to a potential u, we have the corresponding vector-force. From a velocity-potential we obtain the velocity of the fluid element at p ; and from the temperature of a conducting solid we obtain the temperature 373 gradient in the direction of the flux of heat. Finally, whatever series of surfaces is represented by u = G, the vector Vu is the normal at the point p, and its length is inversely as the normal distance at that point between two con secutive surfaces of the series. Hence it is evident that S . dpVa = du, or, as it may be written, the left-hand member therefore expresses total differentiation in virtue of any arbitrary, but small, displacement dp. These results have been already given above, but they were not obtained in such a direct manner. Many very curious and useful transformations may easily be derived (see Ex. 34, Chap. XI.) from the assumption da = (frdp, or / = SaV . cr, where the constituents of * are known functions of p. For instance, if we write _ . d . d j d v - 3f + *i + *3r where a- = i% + jrj + k, we find at once V = Vff , or V^ = ^ ^V ; a formula which contains the whole basis of the theory of the change of independent variables from x, y, z to f, 77, f, or vice versa. The reader may easily develop this application. Its primary interest is, of course, purely mathematical : but it has most important uses in applied mathematics. Our limits, however, do not permit us to reach the regions of its special physical usefulness.

481. To interpret the operator V. aV, let us apply it to a potential function u. Then we easily see that u may be taken under the vector sign, and the expression denotes the vector-couple due to the force at p about a point whose relative vector is a. 374 Again, if cr be any vector function of p, we have by ordinary quaternion operations F(aV) . o- = 8 . a FVo- + aVr - Va7. The meaning of the third term (in which it is of course understood that V operates on a alone) is obvious from what precedes. The other terms were explained in § 384.

J. Applications of V to Line-, Surface-, and Volume-Integrals.

482. In what follows we have constantly to deal with integrals extended over a closed surface, compared with others taken through the space enclosed by such a surface ; or with integrals over a limited surface, compared with others taken round its bounding curve. The notation employed is as follows. If Q per unit of length, of surface, or of volume, at the point p, Q being any quaternion, be the quantity to be summed, these sums will be denoted by ffQds and fflQds, when comparing integrals over a closed surface with others through the enclosed space ; and by ffQds and fQTdp, when comparing integrals over an unclosed surface with others round its boundary. No ambiguity is likely to arise from the double use of ffQds, for its meaning in any case will be obvious from the integral with which it is compared. What follows is mainly from Trans. R. S. E. 1869—70. See also Proc. R. 8. E. 1862—3.

483. We have shewn in § 384 that, if a- be the vector displace ment of a point originally situated at then S.Va . expresses the increase of density of aggregation of the points of the system caused by the displacement.

484. Suppose, now, space to be uniformly filled with points, and a closed surface 5 to be drawn, through which the points can freely move when displaced. 375 Then it is clear that the increase of number of points within the space S, caused by a displacement, may be obtained by either of two processes by taking account of the increase of density at all points within 2, or by estimating the excess of those which pass inwards through the surface over those which pass outwards. These are the principles usually employed (for a mere element of volume) in forming the so-called Equation of Continuity. Let v be the normal to 2 at the point p, drawn outwards, then we have at once (by equating the two different expressions of the same quantity above explained) the equation which is our fundamental equation so long as we deal with triple integrals. [It will be shewn later (§ 500) that the corresponding relation between the single and the double integral can be deduced directly from this.] As a first and very simple example of its use, let p be written for o: It becomes i.e. the volume of any closed space is the sum of the elements of area of its surface, each multiplied by one-third of the perpendicular from the origin on its plane.

485. Next, suppose a to represent the vector force exerted upon a unit particle at p (of ordinary matter, electricity, or magnetism) by any distribution of attracting matter, electricity, or magnetism partly outside, partly inside 2. Then, if P be the potential at p, and if r be the density of the attracting matter, &c., at p, by Poisson’s extension of Laplace’s equation. Substituting in the fundamental equation, we have 47r ///rcfe = 4-rf = ffS . VP Uvds, where M denotes the whole quantity of matter, &c., inside X. This is a well-known theorem.

486. Let P and Px be any scalar functions of p, we can of course find the distribution of matter, &c., requisite to make either 376 of them the potential at p ; for, if the necessary densities be r and r l respectively, we have as before Now V (PVPJ = VPVPl Hence, if in the formula of § 484 we put we obtain JJJS . VPVP^9 = - ///PV2P^9 + ffPS . VPl Uvds, = _ fffP^Pds + JJPfi . VP Uvds, which are the common forms of Green’s Theorem. Sir W. Thomson’s extension of it follows at once from the same proof.

487. If Pl be a many-valued function, but VP1 single-valued, and if 2 be a multiply-connected* space, the above expressions require a modification which was first shewn to be necessary by Helmholtz, and first supplied by Thomson. For simplicity, suppose 2) to be doubly-connected (as a ring or endless rod, whether knotted or not). Then if it be cut through by a surface s, it will become simply-connected, but the surface-integrals have to be increased by terms depending upon the portions just added to the whole surface. In the first form of Green’s Theorem, just given, the only term altered is the last : and it is obvious that if pl be the increase of P after a complete circuit of the ring, the portion to be added to the right-hand side of the equation is pJ/S.VPUvds, taken over the cutting surface only. A similar modification is easily seen to be produced by each additional complexity in the space 2.

488. The immediate consequences of Green’s theorem are well known, so that I take only a few examples. Let P and P1 be the potentials of one and the same distribution of matter, and let none of it be within S. Then we have 2 ds = J/PS . VP Uvds, * Called by Helmholtz, after Biemann, mehrfach susammenhangend. In translating Helmholtz’s paper (Phil. Mag. 1867) I used the above as an English equivalent. Sir W. Thomson in his great paper on Vortex Motion (Trans. R. S E. 1868) uses the expression "multiply-continuous." 377 so that if VP is zero all over the surface of 2, it is zero all through the interior, i.e., the potential is constant inside 2. If P be the velocity-potential in the irrotational motion of an incompressible fluid, this equation shews that there can be no such motion of the fluid unless there is a normal motion at some part of the bounding surface, so long at least as 2 is simply-connected. Again, if 2 is an equipotential surface, ///(VP/ ds = PJfti . V PUvds = P///V 2Pd? by the fundamental theorem. But there is by hypothesis no matter inside 2, so this shews that the potential is constant throughout the interior. Thus there can be no equipotential surface, not including some of the attracting matter, within which the potential can change. Thus it cannot have a maximum or minimum value at points unoccupied by matter.

489. Again, in an isotropic body whose thermal conductivity does not vary with temperature, the equation of heat-conduction is S+-W-0, dt c where (for the moment) k and c represent as usual the conductivity, and the water equivalent of unit volume. The surface condition (assuming Newton’s Law of Cooling) is Assuming, after Fourier, that a particular integral is = e~ mt u, we have V2w-w = 0, * Let um be a particular integral of the first of these linear differential equations. Substitute it for u in the second ; and we obtain (with the aid of the equation of the bounding surface) a scalar equation giving the admissible values of m. Suppose the distribution of temperature when = to be given ; it may be expressed linearly in terms of the various values of u, thus w = For if uiy u^ be any two of these particular integrals, we have by Green’s Theorem, and the differential equations, 378 Hence, unless mA = m2 , we have Ofwk~Q. Thus we have Am fffu*md: = ffjwumds. Am being thus found, we have generally v = I,Ame-""um.

490. If, in the fundamental theorem, we suppose o- = Vr, which imposes the condition that i.e., that the cr displacement is effected without condensation, it becomes ffS . Vr Uvds = fffSV 2rck = 0. Suppose any closed curve to be traced on the surface X, dividing it into two parts. This equation shews that the surface-integral is the same for both parts, the difference of sign being due to the fact that the normal is drawn in opposite directions on the two parts. Hence we see that, with the above limitation of the value of cr, the double integral is the same for all surfaces bounded by a given closed curve. It must therefore be expressible by a single integral taken round the curve. The value of this integral will presently be determined (§ 495).

491. The theorem of § 485 may be written !ffV 2Pck =JfSUvVPd8=ffS(UvV) Pds. From this we conclude at once that if (which may, of course, represent any vector whatever) we have or, if This gives us the means of representing, by a surface-integral, a vector-integral taken through a definite space. We have already seen how to do the same for a scalar-integral so that we can now express in this way, subject, however, to an ambiguity presently to be mentioned, the general integral 379 where q is any quaternion whatever. It is evident that it is only in certain classes of cases that we can expect a perfectly definite expression of such a volume-integral in terms of a surface-integral.

492. In the above formula for a vector-integral there may present itself an ambiguity introduced by the inverse operation to which we must devote a few words. The assumption VV = T is tantamount to saying that, if the constituents of a- are the potentials of certain distributions of matter, &c., those of r are the corresponding densities each multiplied by 4?r. If, therefore, r be given throughout the space enclosed by S, cr is given by this equation so far only as it depends upon the distribution within 2, and must be completed by an arbitrary vector depending on three potentials of mutually independent distributions exterior to . But, if a- be given, r is perfectly definite ; and as Vo- = VV, the value of V 1 is also completely defined. These remarks must be carefully attended to in using the theorem above : since they involve as particular cases of their application many curious theorems in Fluid Motion, &c.

493. As a very special case, the equation of course gives Vcr = u, a scalar. Now, if v be the potential of a distribution whose density is u, we have We know that when u is assigned this equation gives one, and but one, definite value for v. We have in fact, by the definition of a potential, where the integration (confined to u^ and pt) extends to all space in which u differs from zero. 380 Thus there is no ambiguity in and therefore a = Vv 4-7T is also determinate.

494. This shews the nature of the arbitrary term which must be introduced into the solution of the equation FV7 = T. To solve this equation is (§ 384) to find the displacement of any one of a group of points when the consequent rotation is given. Here SVr = S. VFV7 = VV = 0; so that, omitting the arbitrary term (§ 493), we have W= VT, and each constituent of cr is, as above, determinate. Compare § 503. Thomson* has put the solution in a form which may be written if we understand by /( ) dp integrating the term in dx as if y and z were constants, &c. Bearing this in mind, we have as verification, J27i \V-ri + fV^ dp

495. We now come to relations between the results of integra tion extended over a non-closed surface and round its boundary. Let cr be any vector function of the position of a point. The line-integral whose value we seek as a fundamental theorem is jSadr, where T is the vector of any point in a small closed curve, drawn from a point within it, and in its plane. * Electrostatics and Magnetism, § 521, or Phil. Trans., 1852. 381 Let (7 be the value of a at the origin of r, then o- = a- -S(rV)a-0) so that fSadr =JS.{(r -S (rV) aQ ] dr. But fdr = 0, because the curve is closed ; and (ante, § 467) we have generally fSrVS(7 dr = JV (TTOT - T O fVrdr). Here the integrated part vanishes for a closed circuit, and where ds is the area of the small closed curve, and Uv is a unitvector perpendicular to its plane. Hence fSo- dT = S.VTQ Uv.ds. Now, any finite portion of a surface may be broken up into small elements such as we have just treated, and the sign only of the integral along each portion of a bounding curve is changed when we go round it in the opposite direction. Hence, just as Ampère did with electric currents, substituting for a finite closed circuit a network of an infinite number of infinitely small ones, in each contiguous pair of which the common boundary is described by equal currents in opposite directions, we have for a finite unclosed surface There is no difficulty in extending this result to cases in which the bounding curve consists of detached ovals, or possesses multiple points. This theorem seems to have been first given by Stokes (Smith’s Prize Exam. 1854), in the form = rrj (/ (h dfi\ (da dy\ fd/3 da Jjds ll(-r--r) + m(-r --j L ) + n(-^ --:- ( \dy dz) \dz dxj \dx dy It solves the problem suggested by the result of § 490 above. It will be shewn, however, in a later section that the equation above, though apparently quite different from that of § 484, is merely a particular case of it. [If we recur to the case of an infinitely small area, it is clear that ffS.VrUvds is a maximum when F. Z 382 Hence FVcr is, at every point, perpendicular to a small area for which fSadp is a maximum.]

496. If 7 represent the vector force acting on a particle of matter at p, S . adp represents the work done by it while the particle is displaced along dp, so that the single integral JSrdp of last section, taken with a negative sign, represents the work done during a complete cycle. When this integral vanishes it is evident that, if the path be divided into any two parts, the work spent during the particle’s motion through one part is equal to that gained in the other. Hence the system of forces must be conservative, i.e., must do the same amount of work for all paths having the same extremities. But the equivalent double integral must also vanish. Hence a conservative system is such that whatever be the form of the finite portion of surface of which ds is an element. Hence, as Ver has a fixed value at each point of space, while Uv may be altered at will, we must have or Vo- = scalar. If we call X, Y, Z the component forces parallel to rectangular axes, this extremely simple equation is equivalent to the wellknown conditions dX dY dY dZ dZ dX _ ~7-- ~7 - ^J ~7---7 ^ ~J --7 - - - dy dx dz dy ax dz Returning to the quaternion form, as far less complex, we see that Vcr = scalar = 47rr, suppose, implies that cr = VP, where P is a scalar such that that is, P is the potential of a distribution of matter, magnetism, or statical electricity, of volume-density r. 383 Hence, for a non-closed path, under conservative forces, depending solely on the values of P at the extremities of the path.

497. A vector theorem, which is of great use, and which cor responds to the scalar theorem of § 491, may easily be obtained. Thus, with the notation already employed, Now F ( F. V F. rdr) o- = - S(rV) V. r dr - S (drV) VTTO , and d {S (TV) FO-OT} 8 (rV) F. aQdr + S (drV) FT O T. Subtracting, and omitting the term which is the same at both limits, we have Extended as above to any closed curve, this takes at once the form JF. adp = -ffdsV. (FZTi/V) r. Of course, in many cases of the attempted representation of a quaternion surface-integral by another taken round its bounding curve, we are met by ambiguities as in the case of the spaceintegral, 492 : but their origin, both analytically and physically, is in general obvious.

498. The following short investigation gives, in a complete form, the kernel of the whole of this part of the subject. But § 495 7 have still some interest of their own. If P be any scalar function of p, we have (by the process of § 495, above) fPdr=J{P -S(TV)P9}dT = -fS.rVPQ .dr. But F. V F. rdr = drS .rV-rS. drV, and d (rSrV) = drS .rV + rS. drV. These give fPdr = - } {rSrV - F. JF(rdr) V} P = dsV. UvVPQ . Hence, for a closed curve of any form, we have jPdp=JjdsV.UvVP, from which the theorems of §§ 495, 497 may easily be deduced. 384 Multiply into any constant vector, and we have, by adding three such results fdpa- = ffdsV(UvV) r. [See § 497.] Hence at once (by adding together the corresponding members of the two last equations, and putting fdpq=ffdsV(UvV)q, where q is any quaternion whatever. [For the reason why we have no corresponding formula, with S instead of V in the right-hand member, see remark in [ ] in § 505.]

499. Commencing afresh with the fundamental integral put o- = u and we have f/fS{3Vuds = ffuS/3 Uvds ; from which at once fffVuds = ffuUvds ........................ (1), or fJfVTd*=ffUV .Td8 ..................... (2), which gives JJ/FV Yards = Vf}UvVrrds = ff(rSUv(r - crSUvr) ds. Equation (2) gives a remarkable expression for the surface of a space in terms of a volume-integral. For take T =Uv= UVP, where P = const. is the scalar equation of the closed bounding surface. Then _ jfds = fJUv Uvds = ///V UVPds. (Note that this implies which in itself is remarkable.) Thus the surface of an ellipsoid s the integration being carried on throughout the enclosed space. (Compare § 485.) 385 Again, in (2), putting u^r for T, and taking the scalar, we have ffJSrVUl + w,fifV T ) d* = f whence fff[S(rV) a + aSVr\ ds = ffaSrUvds .............. (3). The sum of (1) and (2) gives, for any quaternion, The final formulae in this, and in the preceding, section give expressions in terms of surface integrals for the volume, and the line, integrals of a quaternion. The latter is perfectly general, but (for a reason pointed out in § 492) the former is definite only when the quaternion has the form Vq.

500. The fundamental form of the Volume and Surface Integral is (as in § 499 (1)) fffVuds = ffUvuds. Apply it to a space consisting of a very thin transverse slice of a cylinder. Let t be the thickness of the slice, A the area of one end, and a a unit-vector perpendicular to the plane of the end. The above equation gives at once where dl is the length of an element of the bounding curve of the section, and the only values of Uv left are parallel to the plane of the section and normal to the bounding curve. If we now put p as the vector of a point in that curve, it is plain that V.aUv=Udp, dl = Tdp, and the expression becomes V(oiV)u.A =fudp. By juxtaposition of an infinite number of these infinitely small directed elements, a (now to be called Uv) being the normal vector of the area A (now to be called ds), we have at once ffV (UvV) uds = fudp, which is the fundamental form of the Surface and Line Integral, as given in § 498. Hence, as stated in § 484, these relations are not independent. In fact, as the first of these expressions can be derived at once from the ordinary equation of "continuity," so the second is merely the particular case corresponding to displacements confined to a T. Q. I. 25 386 given surface. It is left to the student to obtain it, simply and directly, (in the form of § 495) from this consideration. [Note. A remark of some importance must be made here. It may be asked : Why not adopt for the proof of the fundamental theorems of the present subject the obvious Newtonian process (as applied, for instance, in Thomson and Tait’s Natural Philosophy, § 194, or in Clerk-Maxwell’s Electricity, § 591) ? The reply is that, while one great object of the present work is (as far as possible) to banish artifice, and to shew the "perfect naturalness of Quaternions/ the chief merit of the beautiful process alluded to is that it forms one of the most intensely artificial applications of an essentially artificial system. Cartesian and Semi-Cartesian methods may be compared to a primitive telegraphic code, in which the different signals are assigned to the various letters at hap-hazard; Quaternions to the natural system, in which the simplest signals are reserved for the most frequently recurring letters. In the former system some one word, or even sentence, may occasionally be more simply expressed than in the latter : though there can be no doubt as to which system is to be preferred. But, even were it not so, the methods we have adopted in the present case give a truly marvellous insight into the real meaning and " inner nature " of the formulae obtained.]

501. As another example of the important results derived from the simple formulae of § 499, take the following, viz.: ffV. V (a Uv) rds = ffaSr Uvds - JfUvSards, where by (3) and (1) of that section we see that the right-hand member may be written = /// {# (TV) a- + o-SVr - VSar] ds = -///F.F(Vo-)Tfc (4). In this expression the student must remark that V operates on r as well as on a. Had it operated on r alone, we should have inverted the order of V and r, and changed the sign of the whole ; or we might have had recourse to the notation of the end of § 133. This, and similar formulae, are easily applied to find the potential and the vector-force due to various distributions of magnetism. To shew how they are to be introduced, we briefly sketch the mode of expressing the potential of a distribution.

K. Application of the ∇ Integrals to Magnetic &c. Problems.

387 502. Let σ be the vector expressing the direction and intensity of magnetisation, per unit of volume, at the element dς. Then if the magnet be placed in a field of magnetic force whose potential is u, we have for its potential energy

E = −∫∫∫Sσ∇udς
= ∫∫∫uS∇dς − ∫∫∫S∇(uσ)dς
= ∫∫∫uS∇dς − ∫∫uSσUνdς .
This shews at once that the magnetism may be resolved into a volume-density S∇σ, and a surface-density −SσUν. Hence, for a solenoidal distribution,
S∇σ = 0.

What Thomson has called a lamellar distribution (Phil. Trans. 1852), obviously requires that

Sσdρ
be integrable without a factor ; i.e., that
V∇σ = 0.
A complex lamellar distribution requires that the same expression be integrable by the aid of a factor. If this be u, we have at once
V∇(uσ) = 0,
or S . σVσ = 0.
But we easily see that (4) of § 501 may be written
−∫∫V . (VσUν) τdς = −∫∫∫V . τV∇σdς − ∫∫∫V . σ∇τdς + ∫∫∫Sσ∇ . τdς.
Now, if τ = ∇ (1/r),
where r is the distance between any external point and the element dς, the last term on the right is the vector-force exerted by the magnet on a unit-pole placed at the point. The second term on the right vanishes by Laplace’s equation, and the first vanishes as above if the distribution of magnetism be lamellar, thus giving Thomson’s result in the form of a surface integral.
the rest not completed

503. As another instance, let a be the vector of magnetic induction, /3 the vector potential, at any point. Then we know, physically, that

fSpdp=ffSUv* t d8.
25-2

388 But. by the theorem of § 496, we have

t/ J fSI3dp=JfS. UvV/3ds.

Since the boundary and the enclosed surface may be any whatever, we must have

a=FV;
and, as a consequence,
Va = 0.

Hence V/5 = a + u,

= V~la + V"1^ + Vq,
where q is a quaternion satisfying the equations

To interpret the other terms, let

V~lu = o-, so that Wo- = ; and o- = Vv,
where v is a scalar such that
V*v = u.

Thus v is the potential of a distribution w/4-Tr, and can therefore be found without ambiguity when u is given. And of course

V"1^ = Vv
is also found without ambiguity.

Again, as SVa. = 0,

we have Va = 7, suppose.

Hence, for any assumed value of 7, we have

so that this term of the above value of /3 is also found without ambiguity.

The auxiliary quaternion q depends upon potentials of arbitrary distributions wholly outside the space to which the investigation may be limited. [Compare this section with § 494.]

504. An application may be made of similar transformations to Ampère’s Directrice de I action electrodynamique, which, § 458 above, is the vector-integral Vpdp Tp 3 389 where dp is an element of a closed circuit, and the integration extends round the circuit. This may be written so that its value as a surface integral is ffs (UvV) V i ds - Of this the last term vanishes, unless the origin is in, or infinitely near to, the surface over which the double integration extends. The value of the first term is seen (by what precedes) to be the vector-force due to uniform normal magnetisation of the same surface. Thus we see the reason why the Directrice can be expressed in terms of the spherical opening of the circuit, as in §§ 459, 471.

505. The following result is obviously but one of an extensive class of useful transformations. Since we obtain at once from § 484 a curious expression for the gravitation potential of a homogeneous body in terms of a surface-integral. [The right-hand member may be written as ffSUvVTpds] and an examination into its nature shews us why we ought not to expect to have a general expression for ffSUvVPds in terms of a line-integral. It will be excellent practice for the student to make this examination himself. Of course, a more general method presents itself in finding the volume-integral which is equivalent to the last written surface-integral extended to the surface of a closed space.] From this, by differentiation with respect to a, after putting p + a for p, or by expanding in ascending powers of To. (both of which tacitly assume that the origin is external to the space 390 integrated through, i.e., that Tp nowhere vanishes within the limits), we have _-f~r v. u~p v.~u.Up 2 and this, again, involves "^ = II ^SUvUpds.

506. The interpretation of these, and of more complex formulae of a similar kind, leads to many curious theorems in attraction and in potentials. Thus, from (1) of § 499, we have ///**-///^*-n-* which gives the attraction of a mass of density t in terms of the potentials of volume distributions and surface distributions. Putting this becomes [[[Vo-ds _ rrr Up . o-cfc _[[Uv. JJJ Tp ])] Tp*"-}) Tp By putting a = p, and taking the scalar, we recover a formula given above ; and by taking the vector we have This may be easily verified from the formula fPdp=VfSUv.VPds, by remembering that VTp= Up. Again if, in the fundamental integral, we put we have ffj^d-2fjj%= jf tSUvUpds. [It is curious how closely, in fact to a numerical factor of one term pres, this equation resembles what we should get by operating on (1) by S.p, and supposing that we could put p under the signs of integration.]

L. Application of V to the Stress-Function.

391 507. As another application, let us consider briefly the Stress-function in an elastic solid. At any point of a strained body let X be the vector stress per unit of area perpendicular to i, ft and v the same for planes perpendicular to j and k respectively. Then, by considering an indefinitely small tetrahedron, we have for the stress per unit of area perpendicular to a unit-vector co the expression \Sia) + pSjw + vSkw = (pco, so that the stress across any plane is represented by a linear and vector function of the unit normal to the plane. But if we consider the equilibrium, as regards rotation, of an infinitely small parallelepiped whose edges are parallel to i, j, k respectively, we have (supposing there are no molecular couples) or or, supposing V to apply to p alone, This shews (§ 185) that in the present case p is self-conjugate, and thus involves not nine distinct constants but only six.

508. Consider next the equilibrium, as regards translation, of any portion of the solid filling a simply-connected closed space. Let u be the potential of the external forces. Then the condition is obviously where v is the normal vector of the element of surface ds. Here the double integral extends over the whole boundary of the closed space, and the triple integral throughout the whole interior. To reduce this to a form to which the method of § 485 is directly applicable, operate by S . a where a is any constant vector whatever, and we have JfS . /a Uvds + {ffdsSaVu = 392 by taking advantage of the self-conjugateness of $. This may be written (by transforming the surface-integral into a volume-integral) fffk (S . V/ + S . aVu) = 0, and, as the limits of integration may be any whatever, S.V_f_oL + SaVu = ...................... (1). This is the required equation, the indeterminateness of a rendering it equivalent to three scalar conditions. There are various modes of expressing this without the a. Thus, if A be used for V when the constituents of p are considered, we may write Vu = It is easy to see that the right-hand member may be put in either of the equivalent forms or In integrating this expression through a given space, we must remark that V and p are merely temporary symbols of construction, and therefore are not to be looked on as variables in the integral. Instead of transforming the surface-integral, we might have begun by transforming the volume-integral. Thus the first equation of this section gives //((/ + u) Uvds = 0. From this we have at once fJS. Uv ( + u) ads = 0. Thus, by the result of § 490, whatever be a we have .V(( + w)a = 0, which is the condition obtained by the former process. As a verification, it may perhaps be well to shew that from this equation we can get the condition of equilibrium, as regards rotation, of a simply-connected portion of the body, which can be written by inspection as JJF. pl ( Uv) ds + /// VpVuds = 0. This is easily done as follows : (1) gives if, and only if, j satisfy the condition 393 Now this condition is satisfied by a = Yap where a is any constant vector. For 8. (V) Vap = -S.aV(l (V)p = S.aW^= 0, in consequence of the self-conjugateness of (/. Hence fffds (S . 7 Vap + 8 . apVu) = 0, or JfdsS . ap(f Uv + fffdsS . apVu = 0. Multiplying by a, and adding the results obtained by making a in succession each of three rectangular vectors, we obtain the required equation.

509. To find the stress-function in terms of the displacement at each point of an isotropic solid, when the resulting strain is small, we may conveniently apply the approximate method of § 384. As the displacement is supposed to be continuous, the strain in the immediate neighbourhood of any point may be treated as homogeneous. Thus, round each point, there is one series of rectangular parallelepipeds, each of which remains rectangular after the strain. Let a, /3, 7 ; a15 {3V yl ; be unit vectors parallel to their edges before, and after, the strain respectively ; and let ev e 2, e3 be the elongations of unit edges parallel to these lines. We shall not have occasion to determine these quantities, as they will be eliminated after having served to form the requisite equations. Since the solid is isotropic and homogeneous, the stress is perpendicular to each face in the strained parallelepipeds ; and its amount (per unit area) can be expressed as Pl = 2we1 + (c - fn) 2e, c (1), where n and c are, respectively, the rigidity and the resistance to compression. Next, as in § 384, let a be the displacement at p. The strain-function is i^tzr = OT $33-V. a so that at once Se = -SVr; (2), and, if q ( ) q 1 be the operator which turns a into a,, cv we have *r (3). 394 Thus, if (f be the stress-function, we have (as in § 507) = - 2 . P^VB (4). But /r = 2 . (1 + e a ) a so that -\J/ft = 2 . (1 + e x and qty wq 1 = 2.(1 +ej afia.^ (5). By the help of (1), (2), and (5), (4) becomes 00) = Zriqifr toq" 1 2nco (c f??) coSVa ; and, to the degree of approximation employed, (3) shews that this may be written (j)0) = -n (SwV. a + VScoa) - (c - fn) wSVa (G), which is the required expression, the function c/ being obviously self-conjugate. As an example of its use, suppose the strain to be a uniform dilatation. Here o- = ep, and (/) = Zneay + 3 (c fn) ea) = Scew ; denoting traction See, uniform in all directions. If e be negative, there is uniform condensation, and the stress is simply hydrostatic pressure. Again, let a eaSap, which denotes uniform extension in one direction, unaccompanied by transversal displacement. We have (a being a unit vector) (f)a) = 2neaS(oa. + (c fn) ew. Thus along a there is traction (c + jn)e, but in all directions perpendicular to a there is also traction (c-|n)e. Finally, take the displacement a = ea.S/3p. It gives ^o) = - ne (aSco/3 + fiSua) -(c- f?i) ewSaff. This displacement gives a simple shear if the unit vectors a and ft are at right angles to one another, and then (f)a) = ne (a.Sco/3 + fiStoa), which agrees with the well-known results. In particular, it shews that the stress is wholly tangential on planes perpendicular either 395 to a. or to {3 ; and wholly normal on planes equally inclined to them and perpendicular to their plane. The symmetry shews that the stress will not be affected by interchange of the unit vectors, a and /3, in the expression for the displacement.

510. The work done by the stress on any simply connected portion of the solid is obviously because $ (Uv) is the vector force overcome per unit of area on the element ds. [The displacement at any moment may be written XCT ; and, as the stress is always proportional to the strain, the factor xdx has to be integrated from to 1.] This is easily transformed to

511. We may easily obtain the general expression for the work corresponding to a strain in any elastic solid. The physical principles on which we proceed are those explained in Appendix G to Thomson and Tait’s Natural Philosophy. The mode in which they are introduced, however, is entirely different; and a comparison will shew the superiority of the Quaternion notation, alike in compactness and in intelligibility and suggestiveness. If the strain, due to the displacement a, viz. ^T = T SrV . a be a mere rotation, in which case of course no work is stored up by the stress, we have at once S . ^TW^TT= S(i)T for all values of co and r. We may write this as S . a) (i|rS/r - 1) r = Sw^r = 0, where % is (§ 380) a self-conjugate linear and vector function, whose complete value is Xr = - SrV . a - VSro- + V^rVSo-ov The last term of this may, in many cases, be neglected. When the strain is very small, the work (per unit volume) must thus obviously be a homogeneous function, of the second degree, of the various independent values of the expression 396 On account of the self-conjugateness of ^ there are but six such values : viz. Their homogeneous products of the second degree are therefore twenty-one in number, and this is the number of elastic coefficients which must appear in the general expression for the work. In the most general form of the problem these coefficients are to be regarded as given functions of p. At and near any one point of the body, however, we may take i, j, k as the chief vectors of ^ at that point, and then the work for a small element is expressible in terms of the six homogeneous products, of the second degree, of the three quantities SiW SJXJ SkXk- This statement will of course extend to a portion of the body of any size if (whether isotropic or not) it be homogeneous and homogeneously strained. From this follow at once all the elementary properties of homogeneous stress.

M. The Hydrokinetic Equations.

512. As another application, let us form the hydrokinetic equations, on the hypothesis that a perfect fluid is not a molecular assemblage but a continuous medium. Let a be the vector-velocity of a very small part of the fluid at p ; e the density there, taken to be a function of the pressure, p, alone ; i.e. supposing that the fluid is homogeneous when the pressure is the same throughout ; P the potential energy of unit mass at the point p. The equation of " continuity" is to be found by expressing the fact that the increase of mass in a small fixed space is equal to the excess of the fluid which has entered over that which has escaped. If we take the volume of this space as unit, the condition is .................... (1). We may put this, if we please, in the form where 3 expresses total differentiation, or, in other words, that we follow a definite portion of the fluid in its motion. 397 The expression might at once have been written in the form (2) from the comparison of the results of two different methods of representing the rate of increase of density of a small portion of the fluid as it moves along. Both forms reduce to when there is no change of density (§ 384). Similarly, for the rate of increase of the whole momentum within the fixed unit space, we have *1 = - evp- Vp +SJSUvo- . eads ; where the meanings of the first two terms are obvious, and the third is the excess of momentum of the fluid which enters, over that of the fluid which leaves, the unit space. The value of the double integral is, by 49.9 (3), (eer) + eSo-V . o- = o- ^ + eSa-V . o-, by (1). Thus we have, for the equation of motion, or, finally fa i * i ^ / ~~ T ^ () This, in its turn, might have been even more easily obtained by dealing with a small definite portion of the fluid. It is necessary to observe that in what precedes we have tacitly assumed that a is continuous throughout the part of the fluid to which the investigation applies : i.e. that there is neither rupture nor finite sliding.

513. There are many ways of dealing with the equation (3) of fluid motion. We select a few of those which, while of historic interest, best illustrate quaternion methods. We may write (3) as Now we have always F. o-FVo- - SoV . a - V^cr = SaV . a - JV . a2 . 398 Hence, if the motion be irrotational, so that (§ 384) Wo- = 0, the equation becomes But, if w be the velocity-potential, 0- = and we obtain (by substituting this in the first term, and operating on the whole by S . dp) the common form 7 dw , 7 2 ir\ d -T. + d . v= - dQ, where l^2 (= |o- 2 ) is the kinetic energy of unit mass of the fluid. If the fluid be incompressible, we have Laplace’s equation for w, viz. V% = SVa = 0. When there is no velocity-potential, we may adopt Helmholtz’ method. But first note the following quaternion transformation (Proc. R. 8. E. 1869—70) [The expression on the right has many remarkable forms, the finding of which we leave, as an exercise, to the student. For our present purpose it is sufficient to know that its vector part is This premised, operate on (3) by V. V, and we have Hence at once, if the fluid be compressible, = -S.VV . 7 + VVa . SVa- = V. V V. ot But if the fluid be incompressible Either form shews that when the vector-rotation vanishes, its rate of change also vanishes. In other words, those elements of the fluid, which were originally devoid of rotation, remain so during the motion. 399 Thomson’s mode of dealing with (3) is to introduce the integral (5) which he calls the " flow " along the arc from the point a to the point p ; these being points which move with the fluid. Operating on (3) with S . dp, we have as above so that, integrating along any definite line in the fluid from a. to p, we have which gives the rate at which the flow along that line increases, as it swims along with the fluid. If we integrate round a closed curve, the value of df/dt vanishes, because Q is essentially a single-valued function. In this case the quantity / is called the " circulation," and the result is stated in the form that the circulation round any definite path in the fluid retains a constant value. Since the circulation is expressed by the complete integral -}Srdp it can also be expressed by the corresponding double integral -//. UvVads, so that it is only when there is at least one vortex-filament passing through the closed circuit that the circulation can have a finite value.

N. Use of V in connection with Taylor’s Theorem.

514. Since the algebraic operator when applied to any function of x, simply changes x into x + h, it is obvious that if a be a vector not acted on by V = ^. ddx + ;. ad,y-+,d-dz, we have e ~ S(rVf(p) =f(p + _r), 400 whatever function f may be. From this it is easy to deduce Taylor’s theorem in one important quaternion form. If A bear to the constituents of or the same relation as V bears to those of p, and if / and F be any two functions which satisfy the commutative law in multiplication, this theorem takes the curious form of which a particular case is (in Cartesian symbols) The modifications which the general expression undergoes, when / and F are not commutative, are easily seen. If one of these be an inverse function, such as, for instance, may occur in the solution of a linear differential equation, these theorems of course do not give the arbitrary part of the integral, but they often materially aid in the determination of the rest. One of the chief uses of operators such as $aV, and various scalar functions of them, is to derive from 1/Tp the various orders of Spherical Harmonics. This, however, is a very simple matter.

515. But there are among them results which appear startling from the excessively free use made of the separation of symbols. Of these one is quite sufficient to shew their general nature. Let P be any scalar function of p. It is required to find the difference between the value of P at p, and its mean value throughout a very small sphere, of radius r and volume vt which has the extremity of p as centre. This, of course, can be answered at once from the formula of § 485. But the somewhat prolix method we are going to adopt is given for its own sake as a singular piece of analysis, not for the sake of the problem. From what is said above, it is easy to see that we have the following expression for the required result : ;///-* -*** where a is the vector joining the centre of the sphere with the element of volume cfc, and the integration (which relates to r and ds alone) extends through the whole volume of the sphere. 401 Expanding the exponential, we may write this expression in the form 1 + JJJ (SaVf Pd,-&c., higher terms being omitted on account of the smallness of r, the limit of To-. Now, symmetry shews at once that JIM? = 0. Also, whatever constant vector be denoted by a, /// (Scwr) 1 = - a2 /// (OrUay tk. Since the integration extends throughout a sphere, it is obvious that the integral on the right is half of what we may call the moment of inertia of the volume about a diameter. Hence If we now write V for a, as the integration does not refer to V, we have by the foregoing results (neglecting higher powers of r) -**-!**-- jS^. which is the expression given by Clerk-Maxwell *. Although, for simplicity, P has here been supposed a scalar, it is obvious that in the result above it may at once be written as a quaternion.

516. As another illustration, let us apply this process to the finding of the potential of a surface-distribution. If p be the vector of the element ds, where the surface density is fp, the potential at a is Jfdsfp.FT(p-r), F being the potential function, which may have any form whatever. By .the preceding, § 514, this may be transformed into JJdsfp.^FTp; or, far more conveniently for the integration, into * London Math. Soc. Proc., vol. iii, no. 34, 1871. T. Q. I. 26 402 where A depends on the constituents of a in the same manner as V depends on those of p. A still farther simplification may be introduced by using a vector CT O , which is finally to be made zero, along with its corresponding operator A , for the above expression then becomes where p appears in a comparatively manageable form. It is obvious that, so far, our formulae might be made applicable to any distribution. We now restrict them to a superficial one.

517. Integration of this last/orm can always be easily effected in the case of a surface of revolution, the origin being a point in the axis. For the expression, so far as the integration is concerned, can in that case be exhibited as a single integral where F may be any scalar function, and x depends on the cosine of the inclination of p to the axis. And da) a As the interpretation of the general results is a little troublesome, let us take the case of a spherical shell, the origin being the centre and the density unity, which, while simple, sufficiently illustrates the proposed mode of treating the subject. We easily see that in the above simple case, a. being any constant vector whatever, and a being the radius of the sphere, C+a 977V7 ffds es*P = 2-Tra e xTadx =^ (e aTa - e- aTa ). J -a -L Now, it appears that we are at liberty to treat A as a has just been treated. It is necessary, therefore, to find the effects of such operators as TA, earA , &c., which seem to be novel, upon a scalar function of To- ; or , as we may for the present call it. 9F Now TA) 2 F = - tfF = F" +~ , whence it is easy to guess at a particular form of TA. To make sure that it is the only one, assume 403 where f and fare scalar functions of ^ to be found. This gives = ?F" + (ft + ff + ff ) F + (ff + f2 ) F. Comparing, we have From the first, f = + 1, whence the second gives f= + ==. ; the signs of f and f being alike. The third is satisfied identically. That is, finally, ^ =^+ 1. Also, an easy induction shews that Hence wo have at once n / d l 1 J +c- by the help of which we easily arrive at the well-known results. This we leave to the student*.

O. Applications of V in connection with Calculus of Variations.

518. We conclude with a few elementary examples of the use of V in connection with the Calculus of Variations. These depend, for the most part, on the simple relation Let us first consider the expression A=fQTdp, where Tdp is an element of a finite arc along which the integration extends, and the quaternion Q is a function of p, generally a scalar. * Proc.R. S. E.t 1871-2. 26-2 404 To shew the nature of the enquiry, note that if Q be the speed of a unit particle, A will be what is called the Action. If Q be the potential energy per unit length of a chain, A is the total potential energy. Such quantities are known to assume minimum, or at least "stationary", values in various physical processes. We have for the variation of the above quantity BA = f (SQTdp + QSTdp) =f(SQTdp-QS.Udpd8p) = - [QSUdpSp] + f {BQTdp + S.SPd (Q Udp)}, where the portion in square brackets refers to the limits only, and gives the terminal conditions. The remaining portion may easily be put in the form SfSp{d(QUdp)-VQ.Tdp}. If the curve is to be determined by the condition that the variation of A shall vanish, we must have, as Bp may have any direction, d(QUdp)-VQ. or, with the notation of Chap. X, This simple equation shews that (when Q is a scalar) (1) The osculating plane of the sought curve contains the vector VQ. (2) The curvature at any point is inversely as Q, and directly as the component of VQ parallel to the radius of absolute curvature.

519. As a first application, suppose A to represent the Action of a unit particle moving freely under a system of forces which have a potential ; so that Q = Tp, and p* = 2(P-H), where P is the potential, H the energy-constant. These give TpVTp = QVQ = - VP, and Qp = p, so that the equation above becomes simply 405 which is obviously the ordinary equation of motion of a free particle.

520. If we look to the superior limit only, the first expression for SA becomes in the present case If we suppose a variation of the constant H, we get the following term from the unintegrated part tSH. Hence we have at once Hamilton’s equations of Varying Action in the forms VA=p dA and -j^. = t. dJd The first of these gives, by the help of the condition above, the well-known partial differential equation of the first order and second degree.

521. To shew that, if A be any solution whatever of this equation, the vector VA represents the velocity in a free path capable of being described under the action of the given system of forces, we have = -S(VA.V)VA. But ~.VA = -S(p Cit A comparison shews at once that the equality VA=p is consistent with each of these vector equations.

522. Again, if 3 refer to the constants only, d(VA)* = S.VAdVA = -dH by the differential equation. But we have also ^v = t, oli which gives %-(dA)=*-S (/3V) dA = dff. Cit 406 These two expressions for dH again agree in giving and thus shew that the differential coefficients of A with regard to the two constants of integration must, themselves, be constants. We thus have the equations of two surfaces whose intersection determines the path.

523. Let us suppose next that A represents the Time of passage, so that the brachistochrone is required. Here we have the other condition being as in § 519, and we have which may be reduced to the symmetrical form It is very instructive to compare this equation with that of the free path as above, § 519 ; noting how the force VP is, as it were, reflected on the tangent of the path (§ 105). This is the well-known characteristic of such brachistochrones. The application of Hamilton’s method may be easily made, as in the preceding example. (Tait, Trans. R. S. E., 1805.)

524. As a particular case, let us suppose gravity to be the only force, then a constant vector, so that d ._! ,_2 jt p p a=0- The form of this equation suggests the assumption where p and q are scalar constants, and Substituting, we get -pq sec2 qt + ( - /3 2 -pV tan2 qt) = 0, which gives pq = T*$ = p*Fa. 407 Now let p{3~ 1 a. = 7 ; this must be a unit-vector perpendicular to a and /3, so that /? p~ l = . (cos at 7 sin otf), cos g v whence p = cos qt (cos qt + y sin gtf) /3" 1 (which may be verified at once by multiplication). Finally, taking the origin so that the constant of integration may vanish, we have 2/o/S = t + ^- (sin 2qt - 7 cos 2q), which is obviously the equation of a cycloid referred to its vertex. The tangent at the vertex is parallel to/3, and the axis of symmetry to a. The equation, it should be noted, gives the law of description of the path.

525. In the case of a chain hanging under the action of given forces we have, as the quantity whose line-integral is to be a minimum, Q = Pr, where P is the potential, r the mass per unit-length. Here we have also, of course, the length of the chain being given. It is easy to see that this leads, by the method above, to the equation where u is a scalar multiplier.

526. As a simple case, suppose the chain to be uniform. Then we may put ru for u, and divide by r. Suppose farther that gravity is the only force, then P = Sap, VP = - a, and -y- {(Sap + ?/) p\ + a = 0. 408 Differentiating, and operating by Sp, we find or, since p is a unit-vector, --0 ds~" which shews that u is constant, and may therefore be allowed for by change of origin. The curve lies obviously in a plane parallel to a, and its equation is (Sap) 2 + a2 s 2 = const., which is a well-known form of the equation of the common catenary. When the quantity Q of § 518 is a vector or a quaternion, we have simply an equation (like that there given) for each of the constituents.

527. Suppose P and the constituents of cr to be functions which vanish at the bounding surface of a simply-connected space 2, or such at least that either P or the constituents vanish there, the others (or other) not becoming infinite.

Then, by § 484, fJfdsS . V(Pa) = // ds PScr Uv = 0, if the integrals be taken through and over 2. Thus fffdsS.aVP = -fffd;PSV7. By the help of this expression we may easily prove a very remarkable proposition of Thomson (Cam. and Dub. Math. Journal, Jan. 1848, or Reprint of Papers on Electrostatics, § 206). To shew that there is one, and but one, solution of the equation 8V (e*Vu) = 477T where r vanishes at an infinite distance, and e is any real scalar whatever, continuous or discontinuous. Let v be the potential of a distribution of density r, so that and consider the integral 409 That Q may be a minimum as depending on the value of u (which is obviously possible since it cannot be negative, and since it may have any positive value, however large, if only greater than this minimum) we must have = JSQ = - fffdiS . (e 2Vu - Vfl) VBu By the lemma given above this may be written Thus any value of u which satisfies the given equation is such as to make Q a minimum. But there is only one value of u which makes Q a minimum ; for, let Ql be the value of Q when ul = u + w is substituted for this value of u, and we have Q, = - jj/* (* v (u+w)-^vv y = Q - 2///cfc S (e 2 Vu - Vw) Vw - fffds e*(Vw) 2 . The middle term of this expression may, by the proposition at the beginning of this section, be written and therefore vanishes. The last term is essentially positive. Thus if MJ anywhere differ from u (except, of course, by a constant quantity) it cannot make Q a minimum: and therefore u is a unique solution.

Miscellaneous Examples.

1. The expression Fa/3 FyS + Fa7 FS/3 + FaS F/37 denotes a vector. What vector ?
2. If two surfaces intersect along a common line of curvature, they meet at a constant angle.
410 3. By the help of the quaternion formulae of rotation, translate into a new form the solution (given in § 248) of the problem of inscribing in a sphere a closed polygon the directions of whose sides are given.
4. Find the point, the sum of whose distances from any number of given points is a minimum. If p be the sought point, al5 2 , &c. the given points: shew that Give a dynamical illustration of this solution. (Proc. R S. E. 1866-7.)
5. Shew that 8.
6. Express, in terms of the masses, and geocentric vectors of the sun and moon, the sun’s vector disturbing force on the moon, and expand it to terms of the second order; pointing out the magnitudes and directions of the separate components. (Hamilton, Lectures, p. 615.)
7. lfq = r*, shew that 2 dq = 2dr* = %(dr + Kqdrq 1 ) Sq~ l = %(dr + q^drKq) Sq~ l = (drq + Kqdr) q~ l (q + Kq)~ l = (drq + Kqdr) (r + Tr)~ l dr + U^drU-1 q~\qdr+Trdrq~*) _drUq+ Uq ~ ldr _ WT Tq(l+ Ur) 1 + Ur - d.r + V. Vdr ~qq-1 = dr - V q q q = drq~ l + V(Vq~\Vdr} (l + and give geometrical interpretations of these varied expressions for the same quantity. (Ibid. p. 628.)
8. Shew that the equation of motion of a homogeneous solid of revolution about a point in its axis, which is not its centre of gravity, is where II is a constant. (Trans. R. S. E., 1869.)
411 9. Find the point P, such that, if A^ A2 , &c. be any fixed points in space, and m1? m2, &c. positive numerical quantities, S.mAP shall be a minimum. Shew that a closed (gauche) polygon can be constructed whose sides are parallel to PA^ PA^ &c. while their lengths are as mv m2 , &c., respectively. If T2, . mAP is to be a minimum, what is the result ?
10. Form the quaternion condition that the lines joining the middle points of the sides of a closed polygon (plane or gauche) may form a similar polygon. When this condition is satisfied, find the quaternion operator which must be applied to the second polygon to make it similarly situated with the first.
11. Solve the equations in linear and vector functions; -or being given, while $ and ^ are to be found : %+ $ =
12. Put the equation of a Minding line (§ 394) directly into the normal form for a line passing through each of two fixed curves : p = x$t + (1 - x) ^u, where and -v/r are definite vector functions of the arbitrary scalars t, and u, respectively.
13. Shew that are different expressions for the same scalar, and give a number of other forms. (§ 513.) How are the values of these quantities respectively affected, if the V without suffix acts on a^ as well as on 7?
14. If (as in § 511) ITT = SCOT for all values of CD and of r, shew by actual transformation, not by the obvious geometrical reasoning, that we have also
412 15. Shew that, to the third order in Tfi, the couple, due to any closed circuit, on a small magnet, 2/3, whose centre is at the origin, is proportional to .p f / sr/81 US! /3 r v ~ 2 v+ T Simplify, by means of this formula, the quasi-Cartesian investiga tion of § 464.
16. Calculate the value of [Vpp J Tp 3 for a circular current, the origin being taken at any point.
17. Find an approximate formula for the potential, near the centre of the field, when two equal, circular, currents have a common axis, and are at a distance equal to their common radius. (This is v. Helmholtz’ Tangent Galvanometer.)
18. Deduce the various forms of Spherical Harmonics, and their relations, from the results of the application of scalar functions of $aV, and similar operators, to l/Tp.
19. Integrate the differential equations : / \ dq , (a) -?- + aq = 6, (b) where a and b are given quaternions, and ( and ^ given linear and vector functions. (Tait, Proc. E. S. E., 1870-1.)
20. Derive (4) of 92 directly from (3) of 91.
21. Find the successive values of the continued fraction -=(/?) where i and j have their quaternion significations, and x has the values 1, 2, 3, &c. (Hamilton, Lectures, p. 645.) 413 22. If we have ;=(/+)" where c is a given quaternion, find the successive values. For what values of c does u become constant ? (Ibid. p. 652.)
23. Prove that the moment of hydrostatic pressures on the faces of any polyhedron is zero, (a) when the fluid pressure is the same throughout, (6) when it is due to any set of forces which have a potential.
24. What vector is given, in terms of two known vectors, by the relation p-^Ma- +r1 )? Shew that the origin lies on the circle which passes through the extremities of these three vectors.
25. Tait, Trans, and Proc. R S. E., 1870-3. With the notation of §§ 484, 495, prove (a) fffS (aV) rd9 -ffrSot Uvds. (b) If 8 (pV) r = - nr, (n + 3) JJ/rd? = - ffrSp Uvds. (c) With the additional restriction W= 0, ffS . UP {2np + (n + 3) / 2Vj . rds = 0. (d) Express the value of the last integral over a non-closed surface by a line-integral. (e) -JTdp=ffdsS.UvVr, if a- = Udp all round the curve. (/) For any portion of surface whose bounding edge lies wholly on a sphere with the origin as centre whatever be the vector a. (g) JVdpV. a = ffds { UvV2 - S ( UvV)V] r, whatever be cr.
26. Tait, Trans. E. S. E., 1873. Interpret the equation da = uqdpq~ l , 414 and shew that it leads to the following results VV = qVuq-\ V. uq~ l = 0, V2 . u* = 0. Hence shew that the only sets of surfaces which, together, cut space into cubes are planes and their electric images.
27. What problem has its conditions stated in the following six equations, from which f, ij, are to be determined as scalar functions of x} y, z, or of p = ix +jy + kz ? __ . d . d 7 d where V= l djx-+JJ j-+k-r dy dz Shew that they give the farther equations Shew that (with a change of origin) the general solution of these equations may be put in the form where (/ is a self-conjugate linear and vector function, and f, 77, f are to be found respectively from the three values of / at any point by relations similar to those in Ex. 24 to Chapter X.
28. Shew that, if p be a planet’s radius vector, the potential P of masses external to the solar system introduces into the equation of motion a term of the form S (pV)VP. Shew that this is a self-conjugate linear and vector function of p, and that it involves only^ye independent constants. Supposing the undisturbed motion to be circular, find the chief effects which this disturbance can produce.
29. In § 430 above, we have the equations where 2 is neglected. Shew that with the assumptions = qrrr~ l q~ 1 we have /3 = 0, T/3 = l, S{3r = 0, provided w/Sfw - ^ = 0. Hence deduce the behaviour of the 415 Foucault pendulum without the x, y, and f, rj transformations in the text. Apply analogous methods to the problems proposed at the end of § 426 of the text.
30. Hamilton, Bishop Law s Premium Examination, 1862. (a) If OABP be four points of space, whereof the three first are given, and not collinear ; if also OA = a, OB = /3, OP = p ; and if, in the equation a a the characteristic of operation F be replaced by S, the locus of P is a plane. What plane ? (6) In the same general equation, if F be replaced by F, the locus is an indefinite right line. What line ? (c) If F be changed to K, the locus of P is a point. What point ? (d) If F be made = V, the locus is an indefinite half-line, or ray. What ray ? (e) If F be replaced by T, the locus is a sphere. What sphere ? (/) If F be changed to TV, the locus is a cylinder of revolution. What cylinder ? (g) If F be made TVU, the locus is a cone of revolution. What cone ? (h) If SU be substituted for F, the locus is one sheet of such a cone. Of what cone ? and which sheet ? (i) If F be changed to F7, the locus is a pair of rays. Which pair ?
31. Hamilton, Bishop Law s Premium Examination, 1863. (a) The equation expresses that / and p are the vectors of two points p and p , which are conjugate with respect to the sphere p 2 + a2 = 0; or of which one is on the polar plane of the other. (b) Prove by quaternions that if the right line PP7 , connect ing two such points, intersect the sphere, it is cut harmonically thereby. 416 (c) If P be a given external point, the cone of tangents drawn from it is represented by the equation, and the orthogonal cone, concentric with the sphere, by (d) Prove and interpret the equation, T(np-a) = T(p-na), if Tp = Ta. (e) Transform and interpret the equation of the ellipsoid, (/) The equation (V - ij = (t 2 + *2 ) Spp + expresses that p and p are values of conjugate points, with respect to the same ellipsoid. (g) The equation of the ellipsoid may also be thus written, Svp = 1, if O2 - * 2 )V = (i - K)*P + ZiSicp + 2fc8ip. (h) The last equation gives also, (i) With the same signification of v, the differential equa tions of the ellipsoid and its reciprocal become Svdp = 0, Spdv = 0. (j) Eliminate p between the four scalar equations, Sap = a, Sfip = b, Syp = c, Sep = e.
32. Hamilton, Bishop Law s Premium Examination, 1864. (a) Let AJ$v Afl^ . . . AnBn be any given system of posited right lines, the 2n points being all given; and let their vector sum, AB = Afr + A + ...+ AnBn, be a line which does not vanish. Then a point H, and a scalar h, can be determined, which shall satisfy the quaternion equation, namely by assuming any origin 0, and writing, nrr v Q^^l + .. 417 (b) For any assumed point C, let Q =CAl .AlBt + ... + CA..AJ.; then this quaternion sum may be transformed as follows, Qc=Qn + and therefore its tensor is in which AB and CH denote lengths. (c) The least value of this tensor TQC is obtained by placing the point C at H; if then a quaternion be said to be a minimum when its tensor is such, we may write min. Qc=QH =h.AB; so that this minimum of Qc is a vector. (d) The equation TQC =c = any scalar constant TQH expresses that the locus of the variable point C is a spheric surface, with its centre at the fixed point H, and with a radius r, or CH, such that r.AB=(T? - T$$ = (6 - h* . AB*)* so that H, as being thus the common centre of a series of concentric spheres, determined by the given system of right- lines, may be said to be the Central Point, or simply the Centre, of that system. (e) The equation TVQc = c 1 = any scalar constant TQH represents a right cylinder, of which the radius divided by AB, and of which the axis of revolution is the line, wherefore this last right line, as being the common axis of a series of such right cylinders, may be called the Central Axis of the system. (/) The equation SQc ~ c 2 any scalar constant represents a plane ; and all such planes are parallel to the Central Plane, of which the equation is SQC =Q. T. Q. I. 27 418 (g) Prove that the central axis intersects the central plane perpendicularly, in the central point of the system. (h) When the n given vectors AlBl,...AnBn are parallel, and are therefore proportional to n scalars, bv ... bn , the scalar h and the vector QH vanish ; and the centre H is then determined by the equation or by the expression, where is again an arbitrary origin.
33. Hamilton, Bishop Laws Premium Examination, 1860. (a) The normal at the end of the variable vector p, to the surface of revolution of the sixth dimension, which is represented by the equation or by the system of the two equations, and the tangent to the meridian at that point, are respectively parallel to the two vectors, v = 2 (p CL) tp, and r - 2 (1 - 2$) (p-a) + tp ; so that they intersect the axis a, in points of which the vectors are, respectively, AOL i u \ JL --- -, and ^- (6) If dp be in the same meridian plane as p, then , and dp (c) Under the same condition, (d) The vector of the centre of curvature of the meridian, at the end of the vector p, is, therefore, 3 v 6a-4-$5 419 (e) The expressions in (a) give v 2 = a2 * 2 (1 - t)\ r2 = aY5 (1 - O2 (4 - 1) ; x 2 9 _ , 2 9a2 ,,2 hence (cr p) = 7 a r, and ap = . dt ; 4 4 t the radius of curvature of the meridian is, therefore, R = T(a-p)= 3 2 tTa; and the length of an element of arc of that curve is * ^M. (/) The same expressions give thus the auxiliary scalar is confined between the limits and 4, and we may write t = 2 vers 0, where is a real angle, which varies continuously from to 2?r ; the recent expression for the element of arc becomes, therefore, ds and gives by integration if the arc’s be measured from the point, say F, for which p a, and which is common to all the meridians; and the total periphery of any one such curve is = 127rTa. (g) The value of cr gives 4 (a 2 - a2 ) = 3a2 * (4 - t\ 16 (Fao-) 2 = - aV (4 - $)" ; if, then, we set aside the axis of revolution a, which is crossed by all the normals to the surface (a), the surface of centres of curvature which is touched by all those normals is represented by the equation, 4(c7 2 -a2 ) 3 + 27a2 (Fa7) 2 = ........ .......... (b). (h) The point F is common to the two surfaces (a) and (b), and is a singular point on each of them, being a triple point on (a), and a double point on (b) ; there is also at it an infinitely sharp cusp on (b), which tends to coincide with the axis a, but a determined tangent plane to (a), which is perpendicular to that axis, and to that cusp ; and the point, say F t of which the vector = a, is another and an exactly similar cusp on (b), but does not belong to (a) . 420 (i) Besides the three universally coincident intersections of the surface (a), with any transversal, drawn through its triple point F, in any given direction ft, there are always three other real intersections, of which indeed one coincides with F if the transversal be perpendicular to the axis, and for which the following is a general formula : (j) The point, say F, of which the vector is p = 2a, is a double point of (a), near which that surface has a cusp, which coincides nearly with its tangent cone at that point; and the / IT semi-angle of this cone is = ^ . Auxiliary Equations : f Svp = - eft (1 -t)(l- 20, \28v (P-CL) = a 2? (1 - 1). f SpT = oi*t*(l-t)(4i-t), \28(p-OL)T = aV (1 - t) (4 - 1).
34. A homogeneous function of p, of the wth degree, is changed into the same function of the constant vector, a, by the operator
35. If r be the cube root of the quaternion q, shew that dr =p + ( V. r2 + r Vr) Vq 1 (rp - pr), where p ^r~*dq. (Hamilton, Lectures, p. 629.)
36. Shew from § 509 that the term to be added to VQ of § 512 (3), in consequence of viscosity, is proportional to
37. Shew, from structural considerations, that must be a linear and vector function of Fa/3. Also prove, directly, that its value is
421 .38. The spherical opening subtended at a by the sphere , - ds -^ r0 according as To. a r. Hence shew, without further integration, that (with the same conditions) f f ds 4-Trr 2 1 1 TjT, x = 477T, or , whence, of course, lldsU(pa)_ Oor- lso that ds 4-Trr 4-Trr 2
39. Find also the values of Uvds Uvds the integration extending over the surface of the sphere Tp = r.
40. Shew that the potential at /3, due to mass m at a, is to the potential at /3" 1 , due to mass m at of1 , as 1 : Tfi, provided m : m :: 1 : Ta. Hence, by the results of 38 above, shew that, with a r, ds 4?rr 4?rr2 (p)a)3 ~ (r - a?) T (/3 - a) according as T/3 r, the limits of integration being as before. Obtain these results directly from Green’s Theorem (§ 486) without employing the Electric Image transformation.
41. Shew that the quaternion [jvS ( UvV) rds - jjja-V *rd; (in the notation of § 482) is changed into its own conjugate by interchange of a and r. Express its value (by §§ 133, 486) as a single volume-integral.
422 42. Prove that jjj(o V V + KVa . V r) d? = jfr UvVrds, and thence that
43. Find the distribution of matter on a given closed surface which will produce, in its interior, the same potential as does a given distribution of matter outside it. Hence shew that there is one, and only one, distribution of matter over a surface, which will produce, at each point of it, any arbitrarily assigned potential.