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Chapter II.
Products and Quotients of Vectors.

45. We now come to the consideration of questions in which the Calculus of Quaternions differs entirely from any previous mathematical method ; and here we shall get an idea of what a Quaternion is, and whence it derives its name. These questions are fundamentally involved in the novel use of the symbols of multiplication and division. And the simplest introduction to the subject seems to be the consideration of the quotient, or ratio, of two vectors.

46. If the given vectors be parallel to each other, we have already seen (§22) that either may be expressed as a numerical multiple of the other; the multiplier being simply the ratio of their lengths, taken positively if they have similar currency, negatively if they run opposite ways.

47. If they be not parallel, let OA and OB be drawn parallel and equal to them from any point ; and the question is reduced to finding the value of the ratio of two vectors drawn from the same point. Let us first find upon how many distinct numbers this ratio depends.

We may suppose OA to be changed into OB by the following successive processes.

1st. Increase or diminish the length of OA till it becomes equal to that of OB. For this only one number is required, viz. the ratio of the lengths of the two vectors. As Hamilton remarks, this is a positive, or rather a signless, number.

2nd. Turn OA about O, in the common plane of the two vectors, until its direction coincides with that of OB, and (remembering 32 the effect of the first operation) we see that the two vectors now coincide or become identical. To specify this operation three numbers are required, viz. two angles (such as node and inclination in the case of a planet’s orbit) to fix the plane in which the rotation takes place, and one angle for the amount of this rotation.

Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends upon four distinct numbers, whence the name Quaternion.

A quaternion q is thus defined as expressing a relation between two vectors α, β. By what precedes, the vectors α, β, which serve for the definition of a given quaternion, must be in a given plane, at a given inclination to each other, and with their lengths in a given ratio ; but it is to be noticed that they may be any two such vectors. [Inclination is understood to include sense, or currency, of rotation from α to β.]

The particular case of perpendicularity of the two vectors, where their quotient is a vector perpendicular to their plane, is fully considered below ; §§64, 65, 72, &c.

48. It is obvious that the operations just described may be performed, with the same result, in the opposite order, being perfectly independent of each other. Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decomposed into two factors of which the order is immaterial. The stretching factor, or that which performs the first operation in §47, is called the Tensor, and is denoted by prefixing T to the quaternion considered.

The turning factor, or that corresponding to the second operation in §47, is called the Versor, and is denoted by the letter U prefixed to the quaternion.

49. Thus, if OA = α, OB = β, and if q be the quaternion which changes α to β, we have

β = qα,
which we may write in the form
β / α = q, or β α−1 = q,
if we agree to define that
(β / α) . α = (β α−1) α = β.
33 Here it is to be particularly noticed that we write q before α to signify that α is multiplied by (or operated on by) q, not q multiplied by α.

This remark is of extreme importance in quaternions, for, as we shall soon see, the Commutative Law does not generally apply to the factors of a product.

We have also, by §§47, 48,

q = Tq.Uq = Uq.Tq,
where, as before, Tq depends merely on the relative lengths of α and β, and Uq depends solely on their directions.

Thus, if α1 and β1 and be vectors of unit length parallel to α and β, respectively,

T(β1/α1) = Tβ1/Tα1 = 1, U(β1/α1) = Uβ1/Uα1 = U(β/α),
As will soon be shewn, when α is perpendicular to β, i.e. when the versor of the quotient is quadrantal, it is a unit-vector.

50. We must now carefully notice that the quaternion which is the quotient when β is divided by α in no way depends upon the absolute lengths, or directions, of these vectors. Its value will remain unchanged if we substitute for them any other pair of vectors which

(1) have their lengths in the same ratio,

(2) have their common plane the same or parallel,

and (3) make the same angle with each other.

Thus in the annexed figure

O1B1 /O1A1 = OB/OA
if, and only if,

(1) O1B1/O1A1 = OB/OA

(2) plane AOB parallel to plane A1O1B1,

(3) ∠AOB = ∠A1O1B1

[Equality of angles is understood to include concurrency of rotation. Thus in the annexed figure the rotation about an axis drawn upwards from the plane is negative (or clock-wise) from OA to OB, and also from O1A1 to O1B1.] T. Q. I.

34 It thus appears that if

β = qα, δ = qγ,
the vectors α, β, γ, δ are parallel to one plane, and may be represented (in a highly extended sense) as proportional to one another, thus :—
β : α = δ : γ.

And it is clear from the previous part of this section that this may be written not only in the form

α : β = γ : δ.
but also in either of the following forms :—
γ : α = δ : β.
α : γ = β : δ.

While these proportions are true as equalities of ratios, they do not usually imply equalities of products.

Thus, as the first of these was equivalent to the equation

β _ α = δ _ γ = q, or βα−1 = γδ−1 = q;
the following three imply separately, (see next section)
α _ β = δ _ γ = q−1, γ / α = δ / β = r, α / γ = β / δ = r−1,
or, if we please,
αβ−1 = γδ−1 = q−1, γα−1 = δβ−1 = r, αγ−1 = βδ−1 = r−1;
where r is a new quaternion, which has not necessarily anything (except its plane), in common with q.

But here great caution is requisite, for we are not entitled to conclude from these that

αδ = βγ, &c.

This point will be fully discussed at a later stage. Meanwhile we may merely state that from

α / β = γ / δ, or β / α = δ / γ,
we are entitled to deduce a number of equivalents such as
αβ−1 = γ, or α = γδ−1β, or β−1δ = α−1γ, &c.

51. The Reciprocal of a quaternion q is defined by the equation

(1 / q) q = q−1q = 1 = q (1 / q) = qq−1.
35
Hence if β / α = q, or
we must have β = q = α
For this gives (α / β).β = q−1.qα.
and each member of the equation is evidently equal to α.

Or thus :—

β = qα.

Operate by q−1,

q−1β = α.

Operate on β−1,

q−1 = αβ−1 = α / β.

Or, we may reason thus :— since q changes OA to OB, q−1 must change OB to OA, and is therefore expressed by α/β (§ 49).

The tensor of the reciprocal of a quaternion is therefore the reciprocal of the tensor ; and the versor differs merely by the reversal of its representative angle. The versor, it must be remembered, gives the plane and angle of the turning—it has nothing to do with the extension.

[Remark. In §§ 49—51, above, we had such expressions as β/α = βα−1. We have also met with α−1β. Cayley suggests that this also may be written in the ordinary fractional form by employing the following distinctive notation :—

β/α = βα−1 = β ___ α ,  α−1β = β ___ α .

(It might, perhaps, be even simpler to use the solidus as recommended by Stokes, along with an obviously correlative type :— thus,

β/α = βα−1 = β/α, α−1β = α\β.)

I have found such notations occasionally convenient for private work, but I hesitate to introduce changes unless they are absolutely required. See remarks on this point towards the end of the Preface to the Second Edition reprinted above.] 3—2

36 52. The Conjugate of a quaternion q, written Kq, has the same tensor, plane, and angle, only the angle is taken the reverse way; or the versor of the conjugate is the reciprocal of the versor of the quaternion, or (what comes to the same thing) the versor of the reciprocal.

Thus, if OA, OB, OA′, lie in one plane, and if

OA′ = OA, and ∠A′OB = ∠BOA, we have
OB ___ OA = q, and OB ___ OA′ = conjugate of q = Kq.

By last section we see that

Kq = (Tq)2q−1.
Hence qKq = Kq . q = (Tq)2.

This proposition is obvious, if we recollect that the tensors of q and Kq are equal, and that the versors are such that either annuls the effect of the other ; while the order of their application is indifferent. The joint effect of these factors is therefore merely to multiply twice over by the common tensor.

53. It is evident from the results of § 50 that, if α and β be of equal length, they may be treated as of unit-length so far as their quaternion quotient is concerned. This quotient is therefore a versor (the tensor being unity) and may be represented indifferently by any one of an infinite number of concurrent arcs of given length lying on the circumference of a circle, of which the two vectors are radii. This is of considerable importance in the proofs which follow.

Thus the versor OB ___ OA may be represented in magnitude, plane, and currency of rotation (§ 50) by the arc AB, which may in this extended sense be written A͡B.

And, similarly, the versor OB1 ___ OA1 is represented by A1͡B1 which is equal to (and concurrent with) A͡B if

∠A1OB1 = ∠AOB,
i.e. if the versors are equal, in the quaternion meaning of the word.

37 54. By the aid of this process, when a versor is represented as an arc of a great circle on the unit-sphere, we can easily prove that quaternion multiplication is not generally commutative.

Thus let q be the versor A͡B or OB ___ OA , where O is the centre of the sphere. Take B͡C = A͡B, (which, it must be remembered, makes the points A, B, C, lie in one great circle), then q may also be represented by OC ___ OB .

In the same way any other versor r may be represented by D͡B or B͡E and by OB ___ OD or OE ___ OB .

[The line OB in the figure is definite, and is given by the intersection of the planes of the two versors.]

Now rOD = OB, and qOB = OC.
Hence qrOD = OC,

or qr = OC ___ OD , and may therefore be represented by the arc D͡C of a great circle.

But rq is easily seen to be represented by the arc A͡E.

For qOA = OB, and rOB = OE OE ,
whence rqOA = OE, and rq = OE ___ OA .

Thus the versors rq and qr, though represented by arcs of equal length, are not generally in the same plane and are therefore unequal: unless the planes of q and r coincide.

Remark. We see that we have assumed, or defined, in the above proof, that q . rα = qr . α and r.qα = rq.α the special case when qα, rα, q . rα, and r . qα are all vectors.

55. Obviously C͡B is Kq, B͡D is Kr, and C͡D is K (qr). But C͡D = B͡D . C͡B, as we see by applying both to OC. This gives us the very important theorem

K (qr) = Kr . Kq,
i.e. the conjugate of the product of two versors is the product of their 38 conjugates in inverted order. This will, of course, be extended to any number of factors as soon as we have proved the associative property of multiplication. (§ 58 below.)

56. The propositions just proved are, of course, true of quaternions as well as of versors; for the former involve only an additional numerical factor which has reference to the length merely, and not the direction, of a vector (§ 48), and is therefore commutative with all other factors.

sections 57-etc. not completed

57. Seeing thus that the commutative law does not in general hold in the multiplication of quaternions, let us enquire whether the Associative Law holds generally. That is if p, q, r be three quaternions, have we

p . qr=pq. r ?
This is, of course, obviously true if^, q, r be numerical quantities, or even any of the imaginaries of algebra. But it cannot be considered as a truism for symbols which do not in general give
pq = qp.

We have assumed it, in definition, for the special case when r, qr, and pqr are all vectors. (§ 54.) But we are not entitled to assume any more than is absolutely required to make our definitions complete.

58. In the first place we remark that p, q, and r may be considered as versors only, and therefore represented by arcs of great circles on the unit sphere, for their tensors may obviously (§ 48) be divided out from both sides, being commutative with the versors.

Let AB=p,EI) = CA = q, and FE = r.

Join BC and produce the great circle till it meets EF in H, and make KH=FE= r, and HG = CB =pq (§ 54).

Join GK. Then JtG = HG . KH = pq . r.

39 Join FD and produce it to meet AB in M. Make LM = FD, and MN= AB, and join NL. Then LN= MN .Lbl=p.qr.

Hence to shew that p . qr = pq . r all that is requisite is to prove that LN, and KG, described as above, are equal arcs of the same great circle, since, by the figure, they have evidently similar currency. This is perhaps most easily effected by the help of the fundamental properties of the curves known as Spherical Conies. As they are not usually familiar to students, we make a slight digression for the purpose of proving these fundamental properties ; after Chasles, by whom and Magnus they were discovered. An independent proof of the associative principle will presently be indicated, and in Chapter VIII. we shall employ quaternions to give an independent proof of the theorems now to be established.

59.* DEF. A spherical conic is the curve of intersection of a cone of the second degree with a sphere, the vertex of the cone being the centre of the sphere.

Lemma. If a cone have one series of circular sections, it has another series, and any two circles belonging to different series lie on a sphere. This is easily proved as follows.

Describe a sphere, A, cutting the cone in one circular section, C, and in any other point whatever, and let the side OpP of the cone meet A in p, P ; P being a point in G. Then PO.Op is constant, and, therefore, since P lies in a plane, p lies on a sphere, a, passing through 0. Hence the locus, c, of p is a circle, being the intersection of the two spheres A and a.

Let OqQ be any other side of the cone, q and Q being points in c, C respectively. Then the quadrilateral qQPp is inscribed in a circle (that in which its plane cuts the sphere A) and the exterior 40 angle at p is equal to the interior angle at Q. If OL, OM be the lines in which the plane POQ cuts the cyclic planes (planes through parallel to the two series of circular sections) they are obviously parallel to pq, QP, respectively ; and therefore z LOp = z Opq = z OQP = z MOQ.

Let any third side, OrR, of the cone be drawn, and let the plane OPR cut the cyclic planes in 01, Om respectively. Then, evidently, ^IOL= Z qpr, Zl/0m= and these angles are independent of the position of the points p and P, if Q and R be fixed points.

In the annexed section of the above space-diagram by a sphere whose centre is 0, IL, Mm are the great circles which represent the cyclic planes, PQR is the spherical conic which represents the cone. The point P represents the line OpP, and so with the others. The propositions above may now be stated thus, Arc PL = arc MQ ; and, if Q and R be fixed, Mm and IL are constant arcs whatever be the position of P.

60. The application to § 58 is now obvious. In the figure of that article we have = GB, LM Hence L, (7, G, D are points of a spherical conic whose cyclic planes are those of AB, FE. Hence also KG passes through L, and with LM intercepts on AB an arc equal to AB. That is, it passes through j\ r , or KG and LN are arcs of the same great circle : and they are equal, for G and L are points in the spherical conic.

41 Also, the associative principle holds for any number of quaternion factors. For, obviously, qr . st = qrs . t = &c., &c., since we may consider qr as a single quaternion, and the above proof applies directly.

61. That quaternion addition, and therefore also subtraction, is commutative, it is easy to shew.

For if the planes of two quaternions, q and r, intersect in the line OA, we may take any vector OA in that line, and at once find two others, OB and 00, such that OB = qOA, and CO = rOA. And (q + r)OA**OB+OC=OC+OB=(r + q) OA, since vector addition is commutative (§ 27).

Here it is obvious that (q + r) OA, being the diagonal of the parallelogram on OB, OC, divides the angle between OB and OG in a ratio depending solely on the ratio of the lengths of these lines, i.e. on the ratio of the tensors of q and r. This will be useful to us in the proof of the distributive law, to which we proceed.

62. Quaternion multiplication, and therefore division, is distributive. One simple proof of this depends on the possibility, shortly to be proved, of representing any quaternion as a linear function of three given rectangular unit-vectors. And when the proposition is thus established, the associative principle may readily be deduced from it.

[But Hamilton seems not to have noticed that we may employ for its proof the properties of Spherical Conics already employed 42 in demonstrating the truth of the associative principle. "For continuity we give an outline of the proof by this process.

Let BA, CA represent the versors of q arid r, and be the great circle whose plane is that of p.

Then, if we take as operand the vector OA, it is obvious that U (q + r) will be represented by some such arc as DA where B, D, C are in one great circle; for (q + r) OA is in the same plane as qOA and rOA, and the relative magnitude of the arcs BD and DC depends solely on the tensors of q and r. Produce BA, DA, CA to meet be in b, d, c respectively, and make Eb = BAt Fd = DA, Gc = CA. Also make bj3 = d = cy=p. Then E, F, G, A lie on a spherical conic of which BC and be are the cyclic arcs. And, because b/3 = d8 = cry, (3E, SF, 7$, when produced, meet in a point H which is also on the spherical conic (§ 59*). Let these arcs meet BC in J, L, K respectively. Then we have LH=F8=pU(q Also LJ and KL = CD. And, on comparing the portions of the figure bounded respectively by HKJ and by ACB we see that (when considered with reference to their effects as factors multiplying OH and OA respectively) pU(q 4- r) bears the same relation to pUq and pUr that U(q + r) bears to Uq and Ur. But T(q + r)U(q + r) = q + r=TqUq+TrUr. Hence T (q + r~) .pU(q + r) = Tq .pUq + Tr .pUr; or, since the tensors are mere numbers and commutative with all other factors, In a similar manner it may be proved that (q + r)p = qp + rp. And then it follows at once that (p + q) (r + s) = pr + ps + qr + qs, where, by § 61, the order of the partial products is immaterial.] 43

63. By similar processes to those of § 53 we see that versors, and therefore also quaternions, are subject to the index-law at least so long as m and n are positive integers.

The extension of this property to negative and fractional exponents must be deferred until we have defined a negative or fractional power of a quaternion.

64. We now proceed to the special case of quadrantal versors, from whose properties it is easy to deduce all the foregoing results of this chapter. It was, in fact, these properties whose invention by Hamilton in 1843 led almost intuitively to the establishment of the Quaternion Calculus. We shall content ourselves at present with an assumption, which will be shewn to lead to consistent results ; but at the end of the chapter we shall shew that no other assumption is possible, following for this purpose a very curious quasi-metaphysical speculation of Hamilton.

65. Suppose we have a system of three mutually perpendicular unit-vectors, drawn from one point, which we may call for shortness i, j, k. Suppose also that these are so situated that a positive (i.e. left-handed) rotation through a right angle about i as an axis brings j to coincide with k. Then it is obvious that positive quadrantal rotation about j will make k coincide with i; and, about k, will make i coincide with j.

For definiteness we may suppose i to be drawn eastwards, j northwards, and k upwards. Then it is obvious that a positive (left-handed) rotation about the eastward line (i) brings the north ward line (j) into a vertically upward position (k) ; and so of the others.

66. Now the operator which turns j into k is a quadrantal versor (§ 53) ; and, as its axis is the vector i, we may call it i.

Thus = i, or k = ?j (1). Similarly we may put -k =j, or i =jk (2), and j=, or J = H (3).

[It may be here noticed, merely to shew the symmetry of the system we arc explaining, that if the three mutually perpendicular 44 vectors i, j, k be made to revolve about a line equally inclined to all, so that i is brought to coincide with j, j will then coincide with k, and k with i : and the above equations will still hold good, only (1) will become (2), (2) will become (3), and (3) will become (i).]

67. By the results of § 50 we see that i.e. a southward unit-vector bears the same ratio to an upward unit-vector that the latter does to a northward one ; and therefore we have or -j =ik Similarly and -k =j, or -k=ji. = K, or i = .(4). (5); (6).

G8. By (4) and (1) we have j = ^k = i(ij) (by the assumption in § 54) = i"j.

Hence i 2 = - 1 (7). And in the same way, (5) and (2) give J 2 = -1 (8), and (6) and (3) 2 = -l (9).

Thus, as the directions of i, j, k are perfectly arbitrary, we see that the square of every quadrantal versor is negative unity.

[Though the following proof is in principle exactly the same as the foregoing, it may perhaps be of use to the student, in shewing him precisely the nature as well as the simplicity of the step we have taken.

Let ABA be a semicircle, whose centre is 0, and let OB be perpendicular to AOA . CYR

Then -=- , = q suppose, is a quadrantal UA versor, and is evidently equal to - ; §§ 50, 53.

Hence OA OH OA -^=.= = -=^ OB OA OA

45 69. Having thus found that the squares of i, j, k are each equal to negative unity ; it only remains that we find the values of their products two and two. For, as we shall see, the result is such as to shew that the value of any other combination whatever of i,j, k (as factors of a product) may be deduced from the values of these squares and products.

Now it is obvious that k i ("Us VHf utM ^^ r W T, fv (i.e. the versor which turns a westward unit-vector into an upward one will turn the upward into an eastward unit) ; or k=j(-i) = -ji* ........................ (10).

Now let us operate on the two equal vectors in (10) by the same versor, it and we have ik = i (ji) = iji. But by (4) and (3) ik = -J = -H.

Comparing these equations, we have or, by § 54 (end), and symmetry gives ij = k,\ jk it \ hi =. } (11).

The meaning of these important equations is very simple ; and is, in fact, obvious from our construction in § 54 for the multiplication of versors ; as we see by the annexed figure, where we must remember that i, j, k are quadrantal versors whose planes are at right angles, so that the figure represents a hemisphere divided into quadrantal triangles. [The arrow-heads indicate the direction of each vector arc.]

Thus, to shew that ij = k, we have, being the centre of the sphere, N, E, 8, W the north, east, south, and west, and ^the zenith (as in § 65) ; 60 JOW=OZ, whence ijOW = iOZ=OS = kOW.

* The negative sign, being a mere numerical factor, is evidently commutative with j ; indeed we may, if necessary, easily assure ourselves of the fact that to turn the negative (or reverse) of a vector through a right (or indeed any) angle, is the same thing as to turn the vector through that angle and then reverse it.

46 70. But, by the same figure, iON=OZ, whence jiON =jOZ = OE = - OW= - kON.

71. From this it appears that and similarly kj = i, J- (12) a jj and thus, by comparing (11), jfc = -kj = i, ((11), (12)). ki = ik=j,

These equations, along with ?=f=V = -l ((7), (8), (9)), contain essentially the whole of Quaternions. But it is easy to see that, for the first group, we may substitute the single equation ijk=-l, (13) since from it, by the help of the values of the squares of i,j, k, all the other expressions may be deduced. We may consider it proved in this way, or deduce it afresh from the figure above, thus

kON= OW, jkON= JOW=OZ, ijkON =

72. One most important step remains to be made, to wit the assumption referred to in § 64. We have treated i, j, k simply as quadrantal versors ; and i, j, k as unit-vectors at right angles to each other, and coinciding with the axes of rotation of these versors. But if we collate and compare the equations just proved we have

f=-i. (7) .a li = -l, (9) (* = 1; (11) Ki= k (i) { = -* 02) bi = -k, (5)

with the other similar groups symmetrically derived from them. 47 Now the meanings we have assigned to i, j, k are quite inde pendent of, and not inconsistent with, those assigned to i, j, k. And it is superfluous to use two sets of characters when one will suffice. Hence it appears that i, j, k may be substituted for i, j, k ; in other words, a unit-vector when employed as a factor may be considered as a quadrantal versor whose plane is perpendicular to the vector. (Of course it follows that every vector can be treated as the product of a number and a quadrantal versor.) This is one of the main elements of the singular simplicity of the quaternion calculus.

73. Thus the product, and therefore the quotient, of two perpendicular vectors is a third vector perpendicular to both.

Hence the reciprocal (§ 51) of a vector is a vector which has the opposite direction to that of the vector, arid its length is the reciprocal of the length of the vector.

The conjugate (§ 52) of a vector is simply the vector reversed.

Hence, by § 52, if a be a vector

(Ta) 2 - aKcL = a ( - a) = - a2 .

74. We may now see that every versor may be represented by a power of a unit-vector.

For, if a be any vector perpendicular to i (which is any definite unit-vector), id, = /3, is a vector equal in length to a, but perpendicular to both i and a.

; ^ / i Z CL = - *, *a fa A _/7r i 4a = ift = i*a = a.

Thus, by successive applications of i, a. is turned round i as an axis through successive right angles. Hence it is natural to define i m as a versor which turns any vector perpendicular to i through m right angles in the positive direction of rotation about i as an axis. Here m may have any real value whatever, whole or fractional, for it is easily seen that analogy leads us to interpret a negative value of m as corresponding to rotation in the negative direction.

75. From this again it follows that any quaternion may be expressed as a power of a vector. For the tensor and versor elements of the vector may be so chosen that, when raised to the same power, the one may be the tensor and the other the versor of the given quaternion. The vector must be, of course, perpendicular to the plane of the quaternion.

48 76. And we now see, as an immediate result of the last two sections, that the index-law holds with regard to powers of a quaternion (§ 63).

77. So far as we have yet considered it, a quaternion has been regarded as the product of a tensor and a versor : we are now to consider it as a sum. The easiest method of so analysing it seems to be the following.

Let - - represent any quaternion. OA Draw BG perpendicular to OA, produced if necessary.

Then, § 19, OB = OC + CB.

But, § 22, OC=xOA, where a? is a number, whose sign is the same as that of the cosine of Z A OB.

Also, § 73, since CB is perpendicular to OA,

A w^*- w H* 14^" .

where 7 is a vector perpendicular to OA and CB, i.e. to the plane of the quaternion; and, as the figure is drawn, directed towards the reader.

Hence OB OA OA Thus a quaternion, in general, may be decomposed into the sum of two parts, one numerical, the other a vector. Hamilton calls them the SCALAR, and the VECTOR, and denotes them respectively by the letters S and V prefixed to the expression for the quaternion.

78. Hence q = Sq+ Vq, and if in the above example OB then OB Vq.OA*.

The equation above gives GB-Vq.OA.

* The points are inserted to shew that S and V apply only to q, and not to qOA

49 79. If, in the last figure, we produce BC to D, so as to double its length, and join OD, we have, by § 52, so that OI)=OC+CD = SKq.OA + VKq.OA. Hence OC = SKq.OA, and CD = VKq.OA. Comparing this value of OC with that in last section, we find SKq = Sq, ................................. (1) or the scalar of the conjugate of a quaternion is equal to the scalar of the quaternion.

Again, CD = CB by the figure, and the substitution of their values gives VKq = -Vq, ........................ (2) or the vector of the conjugate of a quaternion is the vector of the quaternion reversed.

We may remark that the results of this section are simple consequences of the fact that the symbols S, V, K are commutative*. Thus SKq = KSq = Sq, since the conjugate of a number is the number itself; and VKq = KVq = - Vq § 73).

Again, it is obvious that, and thence

80. Since any vector whatever may be represented by cci + yj + zk where a?, y, z are numbers (or Scalars), and i, j, k may be any three non-coplanar vectors, §§ 23, 25 though they are usually understood as representing a rectangular system of unit-vectors and

* It is curious to compare the properties of these quaternion symbols with those of the Elective Symbols of Logic, as given in Boole’s wonderful treatise on the Laws of Thought; and to think that the same grand science of mathematical analysis, by processes remarkably similar to each other, reveals to us truths in the science of position far beyond the powers of the geometer, and truths of deductive reasoning to which unaided thought could never have led the logician.
T. Q. T. 4

50 since any scalar may be denoted by w ; we may write, for any quaternion q, the expression q = w + a?i + yj+zk(§ 78).

Here we have the essential dependence on four distinct numbers, from which the quaternion derives its name, exhibited in the most simple form.

And now we see at once that an equation such as ? =? where q = w + otfi + yj + z k, involves, of course, the four equations w = w, x = x, y = y, z = z.

81. We proceed to indicate another mode of proof of the distributive law of multiplication.

We have already defined, or assumed (§ 61), that 7_ff + 7 a a a or /3cr 1 + 7a-1 = (/3 + 7)a- 1 , and have thus been able to understand what is meant by adding two quaternions.

But, writing a for of1 , we see that this involves the equality (13 + 7) a = @OL + 7 ; from which, by taking the conjugates of both sides, we derive oL (ff + y) = OL p + a y (5:). And a combination of these results (putting {3 + y for a! in the latter, for instance) gives = 13/3 + 7/5 + J3y -f 77 by the former. Hence the distributive principle is true in the multiplication of vectors.

It only remains to shew that it is true as to the scalar and vector parts of a quaternion, and then we shall easily attain the general proof.

Now, if a be any scalar, a any vector, and q any quaternion, (a + a) q = aq + aq- For, if /3 be the vector in which the plane of q is intersected by 51 a plane perpendicular to a, we can find other two vectors, 7 and S, one in each of these planes such that 7 13 *=/ *-?

And, of course, a may be written -~ ; so that aq. And the conjugate may be written ?X + ) = ? + ? * (by the former § 55).

Hence, generally, (a + a) (6 + ) = a + a/3 -f- 6a + a/3 ; or, breaking up a and 6 each into the sum of two scalars, and a, fi each into the sum of two vectors,

(a, + a2 + a, + * 2 ) (6, + + , + s ) = (a, + a,) (6, + 6 8) + (a, -f a,) (

(by what precedes, all the factors on the right are distributive, so that we may easily put it in the form) + K + ,) (62 + /3a ) + (a2 + 2 ) ( Putting 0, + ^=^, a2 + 2 = ^, we have (^ + q) (r + s) =pr + ps + qr + ^5.

82. Cayley suggests that the laws of quaternion multiplication may be derived more directly from those of vector multiplication, supposed to be already established. Thus, let a. be the unit vector perpendicular to the vector parts of q and of q . Then let

p = q.*, a = -a.q,

as is evidently permissible, and we have

pa = q . OLOL = q ; ar = act. . q q ,

so that q . q = px . acr = p . cr. The student may easily extend this process. For variety, we shall now for a time forsake the geometrical mode of proof we have hitherto adopted, and deduce some of our 4—2 52 next steps from the analytical expression for a quaternion given in § 80, and the properties of a rectangular system of unit-vectors as in § 71.

We will commence by proving the result of § 77 anew.

83. Let a = xi + yj + zk, ft = x i + yj + zk. Then, because by § 71 every product or quotient of i,jt k is reducible to one of them or to a number, we are entitled to assume q = - = v + + vj + , where o, f, 77, f are numbers. This is the proposition of § 80.

[Of course, with this expression for a quaternion, there is no necessity for a formal proof of such equations as p + (q+r) = (p + q) + r, where the various sums are to be interpreted as in § 61.

All such things become obvious in view of the properties of i,j,k.]

84. But it may be interesting to find co, , 77, f in terms of x, y, z, x, y , z .

We have ft qa, or x i + yj + z k = (a) + %i + rjj + %k) (xi + yj + zk) as we easily see by the expressions for the powers and products of ij k given in ^1- But the student must pay particular attention to the order of the factors, else he is certain to make mistakes.

This (§ 80) resolves itself into the four equations = fx + vjy + z, z = coz + fy - TJX. The three last equations give j/ xx + yy + zz = w ( 2 + if + 2 ) which determines o.

Also we have, from the same three, by the help of the first, f.r + w + ty = : 53 which, combined with the first, gives fy *M~~ yz zy zx xz xy yx and the common value of these three fractions is then easily seen to be 1_ x* + y 2 + zz

It is easy enough to interpret these expressions by means of ordinary coordinate geometry : but a much simpler process will be furnished by quaternions themselves in the next chapter, and, in giving it, we shall refer back to this section.

85. The associative law of multiplication is now to be proved by means of the distributive (§ 81). We leave the proof to the student. He has merely to multiply together the factors w + xi + yj + zk, w + xi + yj + z k, and w" 4- x"i + y"j + z"kt as follows :

First, multiply the third factor by the second, and then multiply the product by the first ; next, multiply the second factor by the first and employ the product to multiply the third: always re membering that the multiplier in any product is placed before the multiplicand. He will find the scalar parts and the coefficients of i,j, k, in these products, respectively equal, each to each.

86. With the same expressions for a, /3, as in section § 83, we have a/3 = (xi + yj 4 zk) (xi 4 yj + zk) = - (ocx +yy + zz) + (yz - zy ) i + (zx - xz )j + (xy - yx ) k. But we have also /3a = (xx 4 yy 4 zz) (yz zy) i (zx xz )j (xy yx) k. The only difference is in the sign of the vector parts. Hence Safi = S/3a, (1) Fa = -F/3a, (2) also a/3 + /3a = 2$a/3, (3) a/3-a = 2Fa#, (4) and, finally, by § 79, a/3 = ^T./3a (5).

87. If a = /3 we have of course (§ 25) */y - ^/yj y77 ~~y/j /y ^ z z , 54 and the formulae of last section become which was anticipated in § 73, where we proved the formula and also, to a certain extent, in § 25.

88. Now let q and r be any quaternions, then S.qr = S.(Sq+Vq) (Sr+Vr), = S . (8q Sr + Sr. Vq + Sq . Vr + VqVr), = SqSr + S. VqVr, since the two middle terms are vectors. Similarly, S.rq = SrSq + S . Vr Vq. Hence, since by (1) of § 86 we have 8. VqVr = S. VrVq, we see that S.qr = S.rq, (1) a formula of considerable importance.

It may easily be extended to any number of quaternions, because, r being arbitrary, we may put for it rs. Thus we have S . qrs = S . rsq, = S . sqr by a second application of the process. In words, we have the theorem the scalar of the product of any number of given quaternions depends only upon the cyclical order in which they are arranged.

89. An important case is that of three factors, each a vector. The formula then becomes But S.OL@y = SOL (S/3y = SOL Vfiy, since a8/3y is a vector, = -aF7/3, by (2) of § 86, = - SOL (Syj3 + Vy/3) = -S. ay$. Hence the scalar of the product of three vectors changes sign when the cyclical order is altered. 55 By the results of §§ 55, 73, 79 we see that, for any number of vectors, we have K. ct/3y ... /x = %/ V@* (the positive sign belonging to the product of an even number of vectors) so that S . j3 . . . (fx = S xt - *

Similarly F. ... x = *F.x .... Thus we may generalize (3) and (4) of § 86 into 2a...^F-a0...tax*...a, 2V. a/3 ... fo = a/3 ... x + %c - , the upper sign still being used when the -number of factors is even.

Other curious propositions connected with this will be given later (some, indeed, will be found in the Examples appended to this chapter), as we wish to develop the really fundamental formulae in as compact a form as possible.

90. By (4) of § 86, Hence 2V.otV/3y = V.a(8y-yl3) (by multiplying both by a, and taking the vector parts of each side) = F (a/3y + flay j3ay - ay{3) (by introducing the null term @ay pay). That is 2 F . aV/3y = F. (a/3 + 0a) 7 - F (/3a7 + /5Fa7 + S*y./3+ Vay . ) (if we notice that F (F 7 . /3) = - F . /3Fa7? by (2) of § 86). Hence F . a F/fy = ySz{3 - jSSya .................. (1), a formula of constant occurrence.

Adding aS{3y to both sides, we get another most valuable formula V.a0y = aSfa-l3Sya + y8oL0 ............ (2); and the form of this shews that we may interchange 7 and a without altering the right-hand member. This gives F . OLj3y = V . yj3a, a formula which may be greatly extended. (See § 89, above.) 56 Another simple mode of establishing (2) is as follows :

K . a/37 = - 7/3a, .-. 2 V . a/37 - a/37 - K . a/37 (by 79 (2)) = a/37 -f 7/3a = a (Py + 7/3) - (7 4- 7) /3 + 7 (a/3 + /3a)

91. We have also

FFa/3FyS = - FFySFa/3 by (2) of § 86 : - S(7 Fa/3 - ySS Fa/3 = SS . a/37 - yS . a/ = - /3SoL FyS + aS/3 1 78 = - /3S . a7S + a

all of these being arrived at by the help of § 90 (1) and of § 89 ; and by treating alternately Fa/3 and FyS as simple vectors.

Equating two of these values, we have

$S . a/37 = aS . /3jS + 08 . 7S + 7$ . a/3S ......... (3),

a very useful formula, expressing any vector whatever in terms of three given vectors. [This, of course, presupposes that a, /3, 7 are not coplanar, § 23. In fact, if they be coplanar, the factor 8. a/37 vanishes, and thus (3) does not give an expression for 8. This will be shewn in § 101 below.]

92. That such an expression as (3) is possible we knew already by § 23. For variety we may seek another expression of a similar character, by a process which differs entirely from that employed in last section.

a, ft, 7 being any three non-coplanar vectors, we may derive from them three others Fa/3, V(B^y Vya. and, as these will not be coplanar, any other vector 8 may be expressed as the sum of the three, each multiplied by some scalar. It is required to find this expression for 8.

Let 8 = # Fa/3 + 7/F/37 + zVy*- Then SyS = xS . yOL/3 = xS . a/37, the terms in y and z going out, because

7 F/37 = S . 7/37 = 80v* = 7 2 S/3 = 0, for 7 2

is (§ 73) a number.

Similarly S/3S = zS . /37a = zS . a/37, and $a = yS . ifty. Thus $8 . a/37 = Fa/3#78 + F/37/Sfa8 + Vy*S/3B ......... (4).

57 93. We conclude the chapter by shewing (as promised in § 64) that the assumption that the product of two parallel vectors is a number, and the product of two perpendicular vectors a third vector perpendicular to both, is not only useful and convenient, but absolutely inevitable, if our system is to deal indifferently with all directions in space. We abridge Hamilton’s reasoning.

Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed ; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them ; for by changing the signs of both factors the product is unchanged, whereas, as the whole system has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be wholly or partly a number, because on inverting one of them the sign of that number ought to change ; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and (from the symmetry of space) ought to leave the number unchanged. Hence the product of two perpendicular vectors must be a vector, and a simple extension of the same reasoning shews that it must be perpendicular to each of the factors. It is easy to carry this farther, but enough has been said to shew the character of the reasoning.

Examples to Chapter II.

1. It is obvious from the properties of polar triangles that any mode of representing versors by the sides of a spherical triangle must have an equivalent statement in which they are represented by angles in the polar triangle.

Shew directly that the product of two versors represented by two angles of a spherical triangle is a third versor represented by the supplement of the remaining angle of the triangle ; and determine the rule which connects the directions in which these angles are to be measured.

58 2. Hence derive another proof that we have not generally pq = qp.
3. Hence shew that the proof of the associative principle, § 57, may be made to depend upon the fact that if from any point of the sphere tangent arcs be drawn to a spherical conic, and also arcs to the foci, the inclination of either tangent arc to one of the focal arcs is equal to that of the other tangent arc to the other focal arc.

4. Prove the formulae

2$ . a/3y = a/37 - 7/3a, 2F.a/37 = a/37 + 7a.
5. Shew that, whatever odd number of vectors be represented by a, j3, 7, &c., we have always F. a/3y$efr = V. rteSypa, &c.

6. Shew that

8 . Fa/3F/37F7a = -(S. a/37) 2 , F. Vot{3V/3yVyoL=Voi{3(ry 2 SoL/3-S/3vSvoi) + ...... , and F ( Fa/3 F . Vj3y Fya) = ((3Say - a/37) 8 . ay.

7. If a, (3, 7 be any vectors at right angles to each other, shew that

(a 3 -f /3 3 -h y 3 ) S . a/37 = a4 F/37 + /3 4 Fa + 7 4 Fa/3. (a" 1"1 + Pn~l + 7 in-1 ) 8 . a/fy = 2" V/3y + /3 2w Fya + 7 2n Fa/3.
8. If a, /3, 7 be non-coplanar vectors, find the relations among the six scalars, x, y, z and f, 77, f, which are implied in the equation XOL + yfi + 27 = f F/37 + 77 Fya + f FayS.
9. If a, /3, 7 be any three non-coplanar vectors, express any fourth vector, S, as a linear function of each of the following sets of three derived vectors. F.yaft F.a^7, F. ya, and F. Fa/3F/37F7a, F. F/37 Fya Fa/3, F. FyaFa/3F/37.
10. Eliminate p from the equations where a, /3, 7, 8 are vectors, and a, 6, c, d scalars.
11. In any quadrilateral, plane or gauche, the sum of the squares of the diagonals is double the sum of the squares of the lines joining the middle points of opposite sides.