and Vẞp are parallel, i. e. p lies in the same plane as a and ẞ, and can therefore be written (§ 24)

p = xa+yẞ,

where x and y are scalars as yet undetermined.

which (§ 40) is the equation of a hyperbola whose asymptotes are in the directions of a and B.

118.] Again, the equation

Γ.Γαβ Γαρ = 0,

though apparently equivalent to three scalar equations, is really equivalent to one only. In fact we see by § 91 that it may be written - aS.aẞp = 0,

whence, if a be not zero, we have

S.aßp=0,

and thus (§ 101) the only condition is that p is coplanar with a, ß. Hence the equation represents the plane in which a and ẞ lie.

119.] Some very curious results are obtained when we extend these processes of interpretation to functions of a quaternion

q= =w+p

instead of functions of a mere vector ρ.

A scalar equation containing such a quaternion, along with quaternion constants, gives, as in last section, the equation of a surface, if we assign a definite value to w. Hence for successive values of w, we have successive surfaces belonging to a system; and thus when w is indeterminate the equation represents not a surface, as before, but a volume, in the sense that the vector of any point within that volume satisfies the equation.

Thus the equation

or

or

represents, for any assigned whose radius is √a2-w2.

(Tp)2 = a2 — w2,

value of w, not greater than a, a sphere Hence the equation is satisfied by the

vector of any point whatever in the volume of a sphere of radius a, whose centre is origin.

Again, by the same kind of investigation,

(T(q-ẞ))2 = a2,

where q = w+p, is easily seen to represent the volume of a sphere of radius a described about the extremity of ẞ as centre.

Also S(q2)=-a2 is the equation of infinite space less the space contained in a sphere of radius a about the origin.

Similar consequences as to the interpretation of vector equations in quaternions may be readily deduced by the reader.

120.] The following transformation is enuntiated without proof by Hamilton (Lectures, p. 587, and Elements, p. 299).

which, if we take the positive sign, requires

or

st = ±1,

-1

t = ±81= + UKs,
±s−1

which is the required transformation.

then

and therefore

[It is to be noticed that there are other results which might have been arrived at by using the negative sign above; some involving an arbitrary unit-vector, others involving the imaginary of ordinary algebra.]

121.] As a final example, we take a transformation of Hamilton's, of great importance in the theory of surfaces of the second order.

Transform the expression

(Sap)2 +(Sẞp)2+(Syp)2

in which a, ß, y are any three mutually rectangular vectors, into the form

2

·T(ip+pk)、
"
k2—12

which involves only two vector-constants, ↳, K.

{T(ip + px)}2 = (ip+pk) (pi+kp) (§§ 52, 55)

= (12 + K2)p2 + (4ркp + рkpl)

But

(Sap)2+(Sẞp)2 + (Syp)2

=

p2 + 4

(x2 - 12) 2

a ̄2(Sap)2 + B−2(SBp)2+v ̄2(Syp)2 = p2

Multiply by 62 and subtract, we get

(§§ 25, 73).

in value to a2 and y2: and that the right side may do so the term in p2 must vanish. This condition gives

To determine p, substitute in the expression for B2, and we find

4(-K)2

=

2

2

= ( p − 1 ) (a2 — 8 2 ) + ( p + 1 ) ( 3 2 — y2)
(x2-12)2 — —

1
= (x2 + 1 1⁄2) (a2 — y2) − 2 (a2 + y2) + 4/8o.

Thus we have proved the possibility of the transformation, and

determined the transforming vectors, K. 122.] By differentiating the equation

we obtain, as will be seen in Chapter IV, the following,

SapSap+SBpSBp' + SypSyp' =

S.(ip+pk) (kp′+p ́1),

where p also may be any vector whatever.

(x2 - 12) 2

This is another very important formula of transformation; and it will be a good exercise for the student to prove its truth by processes analogous to those in last section. We may merely observe, what indeed is obvious, that by putting p p it becomes the formula of last section. And we see that we may write, with the recent values of and κ in terms of a, ß, y, the identity

aSap+BSBp+ySyp =

K

(12+k2)p+2V.ɩpk
(x2 - 12) 2

(1−k)2p + 2 (iSkp+KSip)
(x2-12)2

=

123.] In various quaternion investigations, especially in such as involve imaginary intersections of curves and surfaces, the old imaginary of algebra of course appears. But it is to be particularly

noticed that this expression is analogous to a scalar and not to a vector, and that like real scalars it is commutative in multiplication with all other factors. Thus it appears, by the same proof as in algebra, that any quaternion expression which contains this imaginary can always be broken up into the sum of two parts, one real, the other multiplied by the first power of √-1. Such an expression, viz. q=d+ √=1q′′,

where qand q" are real quaternions, is called a BIQUATERNION. Some little care is requisite in the management of these expressions, but there is no new difficulty. The points to be observed are: first, that any biquaternion can be divided into a real and an imaginary part, the latter being the product of √−1 by a real quaternion; second, that this √1 is commutative with all other quantities in multiplication; third, that if two biquaternions be equal, as

"

q + √=1q′′ = r + √√ = 1r",

we have, as in algebra, q' = r', q′′ = r";

so that an equation between biquaternions involves in general eight equations between scalars. Compare § 80.

124.] We have, obviously, since √−1 is a scalar,

Hence (§ 103)

=

= (Sq′ + √ − 18q′′ +Vq® + √ −1Vq')(Sq′ + √ —18q′′ — Vq' — √ — 1 Vq′′) = (Sq′ + √ —18q′′′′)2 — (Vq′ + √ — 1Vq′′')2,

The only remark which need be made on such formulæ is this, that the tensor of a biquaternion may vanish while both of the component quaternions are finite.