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Chapter VI.
Sketch of the Analytical Theory of Quaternions.
(By Prof. Cayley.)

(a) Expression, Addition, and Multiplication.

By what precedes we are led to an analytical theory of the Quaternion q = w + ix + jy + kz, where the imaginary symbols i, j, k are such that
i2 + j2 + k2= 1, jk = −kj = i, ki = −ik = j, ij = −ji = k.

The Tensor Tq is = √(w2 + x2 + y2 + z2), and

the Versor Uq is = (1/√(w2 + x2 + y2 + z2)) (w + ix + jy + kz),
which, or the quaternion itself when Tq = 1, may be expressed in the form

cos δ − sin δ(ia + jb + kc) where a2 + b2 + c2= 1 ;
such a quaternion is a Unit Quaternion. The squared tensor w2 + x2 + y2 + z2 is called the Norm.

The scalar part Sq is = w, and the vector part Vq, or say a Vector, is = ix + jy + kz. The Length is = √(x2 + y2 + z2), and the quotient = (1/√(x2 + y2 + z2)) (ix + jy + kz), or say a vector ix + jy + kz where x2 + y2 + z2 = 1, is a Unit Vector.

The quaternions w + ix + jy + kz and w − ix − jy − kz are said to be Conjugates, each of the other. Conjugate quaternions have the same norm ; and the product of the conjugate quaternions is the norm of either of them. The conjugate of a quaternion is denoted by q, or Kq.

p.147 Quaternions q = w + ix + jy + kz and q′ = w′ + ix′ + jy′ + kz′ are added by the formula

q + q′ = w + w′ + i(x + x′) + j(y + y′) + k(z + z′)
the operation being commutative and associative.

They are multiplied by the formula

qq′ = ww′ − xx′ − yy′ − zz′
+ i (wx′ + xw′ + yz′ − zy′)
+ j (wy′ + yw′ + zx′ − z′x)
+ k (wz′ + zw′ + xy′ − x′y),
where observe that the norm is
= (w2 + x2 + y2 + z2) (w′2 + x′2 + y′2 + z′2)
the product of the norms of q and q′.

The multiplication is not commutative, q′q ≠ qq′ ; but it is associative, qq′ . q″ = q . q′q″ = qq′q″ , &c. In combination with addition it is distributive, q(q′ + q″) = qq′ + qq″, &c.

(b) Imaginary Quaternions. Nullitats.

The components w, x, y, z of a quaternion are usually real, but they may be imaginary of the form a + b√(−1), where √(−1) is the imaginary of ordinary algebra: we cannot (as in ordinary algebra) represent this by the letter i, but when occasion requires another letter, say θ, may be adopted (the meaning, θ = √(−1), being explained). An imaginary quaternion is thus a quaternion of the form w + θw1 + i(x + θx1) + j(y + θy1) + k(z + θz1), or, what is the same thing, if q, q1 be the real quaternions w + ix + jy + kz, w1 + ix1 + jy1 + kz1, it is a quaternion q + θq1 ; this algebraical imaginary θ = √(−1) is commutative with each of the symbols i, j, k : or, what comes to the same thing, it is not in general necessary to explicitly introduce θ at all, but we work with the quaternion w, x, y, z, in exactly the same way as if w, x, y, z : or, were real values. A quaternion of the above form, q + θq1, was termed by Hamilton a “biquaternion” but it seems preferable to speak of it simply as a quaternion, using the term biquaternion only for a like expression q + θq1, wherein θ is not the √(−1) of ordinary algebra.

It may be noticed that, for an imaginary quaternion, the squared tensor or norm w2 + x2 + y2 + z2 may be = 0 ; when this is so, the quaternion is said to be a “Nullitat” ; the case is one to be separately considered.

102
p.148

(c) Quaternion as a Matrix.

Quaternions have an intimate connection with Matrices. Suppose that θ = √−1, is the √−1 of ordinary algebra, and in place of i, j, k consider the new imaginaries x, y, z, w which are such that

x = (1/2)( 1 − θi), or conversely 1 =  x + w,
y = (1/2)( j − θk), i = θ(x − w),
z = (1/2)( −j − θk), j =  (y − z),
w = (1/2)( 1 − θi), k = θ(y + z);
so that a, b, c, d being scalars, ax + by + cz + dw denotes the imaginary quaternion
(1/2)[a + d + (b − c)j] + (1/2)[(−a + d)i + (−b − c)k]θ.
We obtain for x, y, z, w the laws of combination
x
y
z
w
xy zw
xy00
00xy
zw00
00zw
, that is x2 = x, xy = y, xz = 0, xw = 0 &c.,

and consequently for the product of two linear forms in (x, y, z, w) we have

(ax + by + cz + dw) (a′x + b′y − c′z + d′w)
= (aa′ + bc′)x + (ab′ + bd′)y + (ca′ + dc′)z + (cb′ + dd′)w ;
and this is precisely the form for the product of two matrices, viz. we have
 
a b
c d
  
 
a′ b′
c′ d′
  
=
(a′, b′) (c′, d′)
(a, b)
 “  “ 
 “  “ 
(c, d)
=
 
aa′ + bc′, ab′ + bd′
ca′ + dc′, cb′ + dd′
  
and hence the linear form ax + by + cz + dw, and the matrix
 
a, b
c, d
  
may be regarded as equivalent symbols. This identification was established by the remark and footnote “Peirce’s Linear Associative Algebra”, Amer. Math. Jour. t. 4 (1881), p. 132.

(d) The Quaternion Equation ΣAqB = C.

In ordinary algebra, an equation of the first degree, or linear equation with one unknown quantity x, is merely an equation of the form ax = b, and it gives at once x = a−1 b.

p.149 But the case is very different with quaternions; the general form of a linear equation with one unknown quaternion q is

A1qB1 + A2qB2 +...= C, or say ΣAqB = C,
where C and the several coefficients A and B are given quaternions.

Considering the expression on the left-hand side, and assuming q = w + ix + jy + kz, it is obvious that the expression is in effect of the form

  δ w + α x + β y + γ z
+ i( δ1w + α1x + β1y + γ1z)
+ j( δ2w + α2x + β2y + γ2z)
+ k( δ3w + α3x + β3y + γ3z),
where the coefficients δ, α, β, γ, &c. are given scalar magnitudes : if then this is equal to a given quaternion C, say this is
λ + iλ1 + jλ2 + kλ3,
we have for the determination of w, x, y, z the four equations
  δ w + α x + β y + γ z = λ
+ δ1w + α1x + β1y + γ1z = λ1
+ δ2w + α2x + β2y + γ2z = λ2
+ δ3w + α3x + β3y + γ3z = λ3,
and we thence have w, x, y, z, each of them as a fraction with a given numerator, and with the common denominator
Δ =
 
δ, α, β, γ
δ1,α1, β1,γ1
δ2,α2, β2,γ2
δ3,α3, β3,γ3
  
viz. this is the determinant formed with the coefficients δ, α, β, γ, &c. Of course if Δ = 0, then either the equations are inconsistent, or they reduce themselves to fewer than four independent equations.

The number of these coefficients is = 16, and it is thus clear that, whatever be the number of the terms A1qB1, A2qB2 &c. we only in effect introduce into the equation 16 coefficients. A single term such as A1qB1 may be regarded as containing seven coefficients, for we may without loss of generality write it in the form

g (1 + ia + jb + kc) q (1 + id +je + kf),
and thus we do not obtain the general form of linear equation p.150 by taking a single term A1qB1 (for this contains seven coefficients only) nor by taking two terms A1qB1, A2qB2 (for these contain 14 coefficients only) ; but we do, it would seem, obtain the general form by taking three terms (viz. these contain 21 coefficients, which must in effect reduce themselves to 16) : that is, a form A1qB1 + A2qB2 + A3qB3 (for these contain 14 is, or seems to be, capable of representing the above written quaternion form with any values whatever of the 16 coefficients δ, α, β, γ, &c. But the further theory of this reduction to 16 coefficients is not here considered.

The most simple case of course is that of a single term, say we have AqB = C : here multiplying on the left by A−1 and on the right by B−1, we obtain at once q = B−1CA−1.

(e) The Nivellator, and its Matrix.

In the general case, a solution, equivalent to the foregoing, but differing from it very much in form may be obtained by means of the following considerations.

A symbol of the above form ΣA ( )B, operating upon a quaternion q so as to change it into ΣA (q)B, is termed by Prof. Sylvester a “Nivellator:” it may be represented by a single letter, say we have φ = ΣA (q)B ; the effect of it, as has just been seen, is to convert the components (w, x, y, z), into four linear functions (w1, x1, y1, z1) which may be expressed by the equation

w1, x1, y1, z1 =
 
δ, α, β, γ
δ1,α1, β1,γ1
δ2,α2, β2,γ2
δ3,α3, β3,γ3
  
(w, x, y, z),
or say by the multiplication of (w, x, y, z), by a matrix which may be called the matrix of the nivellator; and the theory of the solution of the linear equation in quaternions thus enters into relation with that of the solution of the linear equation in matrices.

The operation denoted by φ admits of repetition : we have for instance

{A1( )B1 + A2( )B2}2 = A12( )B12 + A1A2( )B1B2 + A2A1( )B2B1 + A22( )B22;
and similarly for more than two terms, and for higher powers.

Considering φ in connexion with its matrix M, we have M(w, x, y, z) for the components of φ2(q), M(w, x, y, z) for p.151 those of φ3(q), and so on. Hence also we have the negative powers φ−1, &c. of the operation φ. The mode in which φ−1 can be calculated will presently appear: but assuming for the moment that it can be calculated, the given equation is φ(q) = C, that is we have q = φ−1(C), the solution of the equation.

A matrix M of any order satisfies identically an equation of the same order : viz. for the foregoing matrix M of the fourth order we have

 
δ − M, α, β, γ
δ1,α1 − M, β1,γ1
δ2,α2, β2 − M,γ2
δ3,α3, β3,γ3 − M
  
viz. this is
M4 − eM3 + fM2 − gM − h = 0,
where h is the before mentioned determinant
 
δ, α, β, γ
δ1,α1, β1,γ1
δ2,α2, β2,γ2
δ3,α3, β3,γ3
  
, say this is, h = Δ.

M, in its operation on the components (w, x, y, z), of q, exactly represents φ in its operation on q : we thus have

φ4 − eφ3 + fφ2 − gφ − h = 0,
viz. this means that operating successively with on the arbitrary quaternion Q we have identically
φ4(Q) − eφ3(Q) + fφ2(Q) − gφ(Q) − h = 0,
where observe that the coefficients e, f, g, h have their foregoing values, calculated by means of the minors of the determinant : but that their values may also be calculated quite independently of this determinant: viz. the equation shews that there is an identical linear relation connecting the values φ4(Q), φ3(Q), φ2(Q), φ(Q) and Q : and from the values (assumed to be known) of these quantities, we can calculate the identical equation which connects them. But in whatever way they are found, the coefficients e, f, g, h are to be regarded as known scalar functions.

Writing in the equation φ−1(Q) in place of Q, we have

φ3(Q) − eφ2(Q) + fφ1(Q) − gQ − hφ−1(Q) = 0,
viz. this equation gives φ−1(Q) as a linear function of Q, φ(Q), φ2(Q) and φ3(Q) : and hence for the arbitrary quaternion Q writing the value C, we have q = φ−1(C) given as a linear function of p.152 C, φ(C), φ2(C) and φ3(C) : we have thus the solution of the given linear equation.

(f) The Vector Equation ΣAρB = C

The theory is similar if, instead of quaternions, we have vectors. As to this observe in the first place that, even if A, q, B are each of them a vector, the product AqB will be in general, not a vector, but a quaternion. Hence in the equation ΣAqB = C, if C and the several coefficients A and B be all of them vectors, the quantity q as determined by this equation will be in general a quaternion : and even if it should come out to be a vector, still in the process of solution it will be necessary to take account, not only of the vector components, but also of the scalar part ; so that there is here no simplification of the foregoing general theory.

But the several coefficients A, B may be vectors so related to each other that the sum ΣAρB, where ρ is an arbitrary vector, is always a vector 1 ;

1 Thus, if A, B are conjugate quaternions, AρB is a vector σ: this is in fact the form which presents itself in the theory of rotation.
and in this case, if C be also a vector, the equation ΣAρB = C, if will determine ρ as a vector : and there is here a material simplification. Writing ρ = ix + jy + kz, then ΣAρB is in effect of the form
i(α1x + β1y + γ1z)
j(α2x + β2y + γ2z)
k(α3x + β3y + γ3z)
viz. we have these three linear functions of (x, y, z) to be equalled to given scalar values λ1, λ1, λ3 and here x, y, z have to be determined by the solution of the three linear equations thus obtained. And for the second form of solution, writing as before φ = ΣA( ) B, then φ is connected with the more simple matrix
M =
 
α1, β1,γ1
α2, β2,γ2
α3, β3,γ3
  
and it thus (instead of a biquadratic equation) satisfies the cubic equation
 
α1 − M, β1,γ1
α2, β2 − M,γ2
α3, β3,γ3 − M
  
= 0,
say
M3 − eM2 + fM − g = 0.
p.153

We have therefore for φ the cubic equation

φ3 − eφ2 + fφ − g = 0,
and thus φ−1(Q) is given as a linear function of Q, φ(Q), φ2(Q), or, what is the same thing, φ−1(C) as a linear function of C, φ(C), φ2(C) : and (this being so) then for the solution of the given equation φ(ρ) = C we have φ−1(C) a given linear function of C, φ(C), φ2(C).

(g) Nullitats.

Simplifications and specialities present themselves in particular cases, for instance in the cases Aq + qB = C, and Aq = qB, which are afterwards considered.

The product of a quaternion into its conjugate is equal to the squared tensor, or norm ; aa = T2a; and thus the reciprocal of a quaternion is equal to the conjugate divided by the norm ; hence if the norm be = 0, or say if the quaternion be a nullitat, there is no reciprocal. In particular, 0, quà quaternion, is a nullitat.

The equation aqb = c, where a, b, c are given quaternions, q the quaternion sought for, is at once solvable ; we have q = a−1cb−1, but the solution fails if a, or b, or each of them, is a nullitat. And when this is so, then whatever be the value of q, we have aqb a nullitat, and thus the equation has no solution unless also c be a nullitat.

If a and c are nullitats, but b is not a nullitat, then the equation gives aq = cb−1, which is of the form aq = c, and similarly if b and c are nullitats but a is not a nullitat, then the equation gives qb = a−1c, which is of the form qb = c: thus the forms to be considered are aq = c, qb = c, and aqb = c, where in the first equation a and c, in the second equation b and c, and in the third equation a, b, c, are nullitats.

The equation aq = c, a and c nullitats, does not in general admit of solution, but when it does so, the solution is indeterminate; viz. if Q be a solution, then Q + aR, (where R is an arbitrary quaternion) is also a solution. Similarly for the equation qb = c, if Q be a solution, then Q + Sb, (S an arbitrary quaternion) is also a solution : and in like manner for the equation aqb = c, if Q be a solution then also Q + aR + Sb, (R, S arbitrary quaternions) is a solution.

p.154

(h) Conditions of Consistency, when some Coefficients are Nullitats.

Consider first the equation aq = c; writing a = a4 + ia1 + ja2 + ka3, (a42 + a12 + a22 + a32 = 0) and c = c4 + ic1 + jc2 + kc3, (c42 + c12 + c22 + c32 = 0) ; also q = w + ix + jy + kz, the equation gives

c4 = a4w − a1x − a2y − a3z,
c3 = a1w + a4x − a3y + a2z,
c2 = a2w + a3x + a4y − a1z,
c1 = a3w − a2x + a1y + a4z,
equations which are only consistent with each other when two of the c’s are determinate linear functions of the other two c’s ; and when this is so, the equations reduce themselves to two independent equations. Thus from the first, second and third equations, multiplying by a1a3 − a2a4, −a4a3 − a1a2, and −a22 − a32, and adding, we obtain
(a1a3 − a2a4)c4 − (a4a3 + a1a2)c1 − (a22 + a32)c2 = 0;
similarly from the first, second and fourth equations, multiplying by −a1a3 − a2a4, −a1a3 + a2a4, −a22 − a32, and adding, we have
−(a1a2 + a3a4)c4 − (a1a3 − a2a4)c1 − (a22 + a32)c3 = 0,
and when these two equations are satisfied, the original equations are equivalent to two independent equations ; so that we have for instance a solution Q = w + ix where c4 = a4w − a1x, c1 = a1w − a4x, that is w = (a4c4 + a1c1) / (a42 + a12), x = −(a1c4 + a4c1) / (a42 + a12); and the general solution is then obtained as above.

The equations connecting the a’s and the c’s may be presented in a variety of different forms, all of them of course equivalent in virtue of the relations a42 + a12 + a22 + a32 = 0, c42 + c12 + c22 + c32 = 0; viz. writing

A1 = a12 + a42 = −a22 − a32, F1 = a2a3 + a1a4, F′1 = a2a3 − a1a4,
A2 = a22 + a42 = −a32 − a12, F2 = a3a1 + a2a4, F′2 = a3a1 − a2a4,
A3 = a32 + a42 = −a12 − a22, F3 = a1a2 + a3a4, F′3 = a1a2 − a3a4,
then the relation between any three of the c’s may be expressed in three different forms, with coefficients out of the sets A1, A2, A3; F1, F2, F3; F′1, F′2, F′3. Obviously the relation between the c’s is satisfied if c = 0: the equation then is aq = 0, satisfied by q = aR, R an arbitrary quaternion. p.155

We have a precisely similar theory for the equation qb = c: any two of the c’s must be determinate linear functions of the other two of them; and we have then only two independent equations for the determination of the w, x, y, z.

In the case of the equation aqb = c (a, b, c all nullitats) the analysis is somewhat more complicated, but the final result is a simple and remarkable one ; from the condition that a, b are nullitats, it follows that ab, aib, ajb, akb are scalar (in general imaginary scalar) multiples of one and the same nullitat, say of ab: the condition to be satisfied by c then is that c shall be a scalar multiple of this same nullitat, say c = λab ; the equation aqb = λab ; has then a solution q = λ, and the general solution is q = λ + aR + bR, where R, S are arbitrary quaternions.

sections i-etc. not completed

(i) The Linear Equations, aq − qb = 0, and aq − qb = c.

The foregoing considerations explain a point which presents itself in regard to the equation aq-qb = 0, (a, b given quaternions, q a quaternion sought for): clearly the equation is not solvable (otherwise than by the value q = 0) unless a condition be satisfied by the given quaternions a, b; but this condition is not (what at first sight it would appear to be) T2a = T2 b. The condition (say ft) may be satisfied although T*a=t= T2 b } and being satisfied, there exists a determinate quaternion q, which must evidently be a nullitat (for from the given equation aq = qb we have (To, - T*b) T*q = 0, that is T2 q = 0). If in addition to the con dition ft we have also Tza T*b = 0, then (as will appear) we have an indeterminate solution q, which is not in general a nullitat.

Take the more general equation aq qb = c: this may be solved by a process (due to Hamilton) as follows: multiplying on the left hand by a and on the right hand by b, we have daq aqb dc, aqb qb* = cb, whence subtracting

aaq (a + a) qb + qb 2 = ac cb,
or since ad, a + d are scalars q {ad (a 4- d) b + 6 2 } = ac cb: viz. this is an equation of the form qB = C (B, C given quaternions), having a solution q = CB~ 1 .

Suppose c = 0, then also (7 = 0; and unless B is a nullitat, the equation qB = (representing the original equation aq = qb), has only the solution g = 0; viz. the condition in order that the p.156 equation aq = qb may have a solution other than q = 0, is B = nullitat, that is aa (a + a) b + 62 = nullitat ; viz. we must have

+ b? + 264 (ib, +j\ + kb3 ) - 6, a - _, - 63 8 = nullitat,
that is
^ + + a, 8 4- a8 8 - 2/. 44 + 64 8 - _/ - 68 8 - 63 2 + 2 (64 - a4) (t 6, +j/_2 4- 63) = nullitat.

The condition ft thus is that is

j(a4 - 64) 8 + 0 + a./ 4- a3 2 - 6 1 3 -68 8 -687+ 4(a4 -64 ) 8 (61 s +6;+68 s)=0,
or, as this may also be written,
( 4 - 6 4) 4 4- 2 ( 4 - 64) 2 (ai 8 + 0 + 3 2 + b: 4 6, 8 + 63 2 ) + + 0 + 0 - 6 t 2 - 6.; - 6 8 8)* = 0.

Writing herein

a + a, 9 + a? + a = ^L 8 , 64 2 + 6 a 2 + 62 2 + 6 3 2 = ^2 ,
the condition is
(a4 - 64) 4 + 2 ( 4 - 64 ) 8 (4* + 2 - a4 8 - _4") + (^L 2 - B2 - a* + 64 2) 2 = 0,
which is easily reduced to
4 (a, - 64) (a4^2 - 6 4 ^1 2) + (A* - BJ = 0,
and, as already noticed, this is different from T2a T2b = 0, that is A*-_ = 0.

If the equation A* B* = Q is satisfied, then the condition 1 reduces itself to a4 6 4 = 0; we then have a = a- 4 -|-a, 6 = a4 + /5, where a, y5 are vectors, and the equation is therefore aq = q/3 where (since A 2 -B\ = ax 2 + a2 2 + ct 3 2 - b* - b* - 63 2 , = 0), the tensors are equal, or we may without loss of generality take a, /3 to be given unit vectors, viz. we have a2 = ] , /3 2 = 1 : and this being so, we obtain at once the solution q = X (a + /3) + //,(! - a/3) (X, IJL, arbitrary scalars): in fact this value gives

aq = X (- 1 + a/3) + p (a + 0) = q@.

If the equation A* B* = Q is satisfied, then the condition 1 Reverting to the general equation oq qb = c, the conjugate of ad(a-\-d)b + b 2 is ad (a + a)b + b\ and we thus obtain the solution

2 {* K- 64) K^2-Ma ) + (^ 2-^2 ) 2 } = (c - cb) [ad -(a + a)b + 6 2 },
p.157 but this solution fails if ad (a + a)b + 6 2 is a imllitat : supposing it to be so, the equation is only solvable when C satisfies the condition which expresses that the equation qB = C is solvable when B, C are nullitats.

The equation aq qb = c, could it is clear be in like manner reduced to the form Aq= C.

(j) The Quadric Equation q2 2aq + b = 0.

We consider the quadric equation (f 2aq + 6 = 0; a and b given quaternions, q the quaternion sought for. The solution which follows is that given by Prof. Sylvester for a quadric equation in binary matrices.

In general if q be any quaternion, = w 4- ix +jy + kz, then (q-w) 2 4- x?+ y*+z* = 0, that is q* - 2qw + n? 4- a? + y 2 + z~ = 0, or say (f 2q (seal, q) + norm q = : viz. this is an identical relation connecting a quaternion with its scalar and its norm.

Writing as above q = w + i_+jy + kzt and t = w* + a? -f?/ 2 + 2 for the norm, we thus have

cf - 2wq + t = 0,
and combining this with the given equation
q*--2aq + b = 0,
we find 2 (a - w) q-(b-t) = 0, that is 2q = (a - w) 1 (b - t), an expression for q in terms of the scalar and norm w, t, and of the known quaternions a and b.

2q as thus determined satisfies the identical equation

(2#) 2 - 2 (2q) seal, {(a - lu) 1 (_-*)} + norm {(a - w)~ l (b - 1)} = 0,
and we have
\-i/7 *\) seal, {(a w) (_ $)! seal, {(a- w) l (b-t)\ =- -, norm (a - w) \-i/i ,M norm (6-^)
norm\(a-w) l (b-t)}= ^ norm (a w)
(a the conjugate of a).

The equation thus becomes

4g 2 norm (a w) ^q (seal, (d w)(b t)} -f norm (b t) = :
this must agree with
(f -2qw + t =0,
p.158 or say the function is = 4X (q z 2qw + t) ; we thus have
norm (a w) X,
seal, (a w}(b t} 2Xw,
norm (b t) - 4X,
three equations for the determination of X, w, t ; and then, w, t being determined, the required value of q is *2q = (a w)~ l (b t) as above.

To develope the solution let the values of a, b, c, f, g, h be denned as follows : viz.

norm (ax + by 4 z) = (a, .b, c, f, g, h$#, y, z) z , viz. writing a = a4 + ia l +ja2 4 kaa, b = b4 + i\ 4 j62 4 kbs,
then this equation is
(a4x + by 4 zf + (av x 4 6 ty)* + (yc + 6 22/) 2 + (a3 + 632/) 2 = (a, b, c, f, g, hja?, y, 0) 2 ,
that is, a, b, c, f, g, h denote as follows
a = a? + a* + a* + a3 2 , f = 6 4 ,
b = 64 2 + ^2 +62 2 + 63 2 , g=a4,
c = 1, h = a464 + aA 4- aA + aA

We then have

norm (a w) = (a4 - -w;) 2 + a^ 4- 2 8 4 as 2 ,
seal, (a -w)(b-t) = (a4 - w) (64 - 4 a l b l 4 a2 68 4 as 68 ,
norm (b - t) = (64 - *) a + 6 : 2 4 6, 2 4 _3 2 ,
or expressing these in terms of (a, b, c, f, g, h) the foregoing three equations become
a - 2gw 4 cw* = X,
h g fw + ctw = Z\w,
b-2ft 4c^2 =4M,
where c (introduced only for greater symmetry) is = 1.

Writing moreover A, B, C, F, G, #=bc-f2 , ca-g2 , ab-h2 , gh - af, hf - bg, fg - ch, and K= abc - af2 - bg 2 - ch2 + 2fgh ; also in place of w, t introducing into the equations n =w- g, and v = t f, the equations become

u* + B=\,
uv -H = 2\(u + g),
tf+A= 4X (v 4 f).

p.159

We deduce u* = X - B,
and we thence obtain, to determine X, the cubic equation
(X - B) (4X 2 + 4Xf- A) - (2Xg + H)2 = 0,
viz. this is
4X3 4- 4X2 (f - a) + X {- be + f 2 + 4 (gh - af)} 4- c (abc - af2 - bg 2 - ch2 + 2fgh) = 0,
that is, 4X3 + 4X2 (f - a) + X (- A + 4F) + K = 0,
and, X being determined by this equation, then
and then w = u + g, t = v+f; consequently
2q = (a-g-u)-1 (b-f-v).

Write for a moment a g u = , then

( + 2u) = (a-gy-u2 = az -2ag + _-B-u\ =-X
(since a = g + iat +ja9 + kav a = g z + a1 2 + a* + a* and thus the identical equation for a is aJ 2ag + a = 0): that is O2+2^@+X= 0, or X 1 = - ( -f 2u) = (a-g + u) ; that is 1 , = (a - g - u)~\ = -(a g+u) , and the value of q is 2q= - (ag+u)(bfv\ X A or say it is
where X is determined by the cubic equation, and u is = + J\ B; we have thus six roots of the given quadric equation q 2 2aq + 6 = 0.