we see that the two vectors now coincide or become identical. To specify this operation three more 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.

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 = a, OB= ß, and if q be the quaternion which changes a to B, we have

Here it is to be particularly noticed that we write q before a to signify that a is multiplied by q, not q multiplied by a.

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 a and B, and Uq depends solely on their directions.

Thus, if a, and ẞ, be vectors of unit length parallel to a and ẞ

As will soon be shewn, when a is perpendicular to ß, the versor of the quotient is quadrantal, i. e. it is a unit-vector.

50.] We must now carefully notice that the quaternion which is the quotient when ẞ is divided by a 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

and

(1) have their lengths in the same ratio,

(2) have their common plane the same or parallel, (3) make the same angle with each other. Thus in the annexed figure

(2) plane AOB parallel to plane à ̧О1В1,
(3) LAOB = LA101 B1.

[Equality of angles is understood to include similarity in direction. Thus the rotation about an upward axis is negative (or right-handed) from OA to OB, and also from O̟11 to О1 В1.]

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

and each member of the equation is evidently equal to a.

Or, we may reason thus, q changes OA to OB, q1 must therefore

a

change OB to OA, and is therefore expressed by (§ 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.

52.] The Conjugate of a quaternion q, written Kq, has the same tensor, plane, and angle, only the angle is taken the reverse way. Thus, if OA, OB, OA', lie in one plane, and if

OA' OA, and LAOB LAOB, we have

=

OB

OA

=

OB
= q, and = conjugate of q =
Ο Α'

By last section we see that

A

A

Hence

-1

Kq = (Tg)2 q ̄1,

qKq=Kq.q= (Tq)2.

Kq.

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. 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 a and ß be of equal length, their quaternion quotient becomes a versor (the tensor being unity) and may be represented indifferently by any one of an infinite number of 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.

i.e. if the versors are equal, in the quaternion meaning of the word.

.

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.

In the same way any other versor r

may be represented by DB or BE and by

The line OB in the figure is definite, and is given by the intersection of the planes of the two versors; O being the centre of the unit-sphere.

a great circle.

, and may therefore be represented by the arc DC of

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

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.

Calling OA a, we see that we have assumed, or defined, in the above proof, that q.ra qr.a and r.qa rq.a when qa, ra, q.ra, and r.qa are all vectors.

=

=

55.] Obviously CB is Kq, BD is Kr, and CD is K (gr). But CD = BD.CB, which gives us the very important theorem

i.e. the conjugate of the product of two quaternions is the product of their conjugates in inverted order.

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).

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. That is, if p, q, r be three quaternions, have we

p.qr = pq.r?

This is, of course, obviously true if p, 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.

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, for their tensors may obviously (§ 48) be divided out from both sides, being commutative with the versors.

Let AB = p, ED = 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).

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 are evidently measured in the same direction. This is perhaps most easily effected by the help of the fundamental properties of the curves known as Spherical Conics. 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 VII 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.