Thus we have proved the possibility of the transformation, and

determined the transforming vectors ↳, K.

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

Sap Sap' + SBp SBp' + Syp Syp' =

S. (ip+pk) (kp' + p ́c)

where p' also may be any vector whatever.

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 x in terms of a, ß, y, the identity

aSap+BSBp+ySyp=

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

M

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.

where q and 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 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′′' = "";

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

124. We have, obviously, since √-1 is a scalar, 8 (q + √ − 1 q') = Sq′ + √ − 1 Sq′′,

Hence (§ 103)

V (q + √ − 1q′′') =Vq + √ −1Vq'.

2

{T(q' + √ −1q')}2 = (Sq′)2 — (Sq′)' — (Vq)2 + (Vq′′′)2

+ 2 √√ −1 { Sq Sq' — S.VqVq'},

= (Tg)2 — (Tq′′)2 + 2 √ — 18.qKq'.

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.

is the general form of a biquaternion whose tensor is zero.

125. More generally we have, q, r, q, r being any four real and non-evanescent quaternions,

(q+√=1g')(r + √=−1r) = gr−qr + √ — 1(qr2 + qʻr).

That this product may vanish we must have

where a is some unit-vector.

And the two equations now agree in giving

so that we have the biquaternion factors in the form

g'(a+1) and (a-√1);

and their product is

-g(a+√1)(a−√—1)r',

which, of course, vanishes.

[A somewhat simpler investigation of the same proposition may be obtained by writing the biquaternions as

(q-q+1) and (+√1)r′,

or q('+1)
-1) and (+√√−1)r',

and showing that

'-"=a, where Ta = 1.]

From this it appears that if the product of two bivectors

p+o√1 and po√1

is zero, we must have

where a may be any vector whatever. But this result is still more easily obtained by means of a direct process.

126. It may be well to observe here (as we intend to avail ourselves of it in the succeeding Chapters) that certain abbreviated forms of expression may be used when they are not liable to confuse, or lead to error. Thus we may write T'q for (Tq)",

just as we write

cos' for (cos 0)",

although the true meanings of these expressions are

T(Ta) and cos (cos 0).

The former is justifiable, as T(Ta) = Ta, and therefore T'a is not required to signify the second tensor (or tensor of the tensor) of a. But the trigonometrical usage is quite inde

fensible.

Similarly we may write

Saq for (Sq), &c.

but it may be advisable not to use

Sq

as the equivalent of either of those just written; inasmuch as it might be confounded with the (generally) different quantity

S.q2 or S(q2),

although this is rarely written without the point or the brackets.

127. The beginner may expect to be a little puzzled with this aspect of the notation at first; but, as he learns more of the subject, he will soon see clearly the distinction between such an expression as

S.VaẞVBY,

where we may omit at pleasure either the point or the first V without altering the value, and the very different one

Saß.VBY,

which admits of no such changes, without altering its value. All these simplifications of notation are, in fact, merely examples of the transformations of quaternion expressions to which part of this Chapter has been devoted. Thus, to take a very simple example, we easily see that

S.VaẞVBy =

SVaẞVẞy = S.aẞVßy = Sa V.ẞVẞy=—SaV.(Vẞy) B =SaV.(Vyẞ) B= S.aV(YB)B = S.V(yẞ) ẞa = SVуßVßa = S.yẞVẞa = &c., &c.

Ba

The above group does not nearly exhaust the list of even the simpler ways of expressing the given quantity. We recommend it to the careful study of the reader.

EXAMPLES TO CHAPTER III.

1. Investigate, by quaternions, the requisite formulæ for changing from any set of coordinate axes to another; and derive from your general result, and also from special investigations, the common expressions for the following cases :

(a.) Rectangular axes turned about through any angle.