SECTION 2

Coplanar Quaternions

64°. The multiplication of diplanar quaternions is not commutative the multiplication of coplanar quaternions is commutative.

For, if two quaternions, q and r, are coplanar, Vq and Vrare parallel; and, consequently, V. VqVr=o=V. VrVq, 23°. Therefore the two equations of 60° (4) are equal, and

qr = rq.

65°. Hence, any quaternion, its reciprocal, its opposite, its conjugate, and any power of the quaternion, all of which are coplanar, are commutative; or,

qKq=Kq . q ; q ̄1 ( − q) = − q · q ̄1 ; Kq . q”= q”Kq, &c.,

SECTION 3

Right Quaternions

&c.

66°. Let v1, v2 be any two right quaternions, with axes ← and ʼn respectively. Then,

η

= Ax. v1 = ← ;

= 0; Vv1: = v1 ; Uv1
Κυι
Kv1 = KVv1 = − Vv1 = − v1 j

with corresponding values for v2.

Consequently, equations (1) and (2) of 60° become, v1v2=S. V v1Vv2+V. Vv1Vv2=Sv1v2+V v1v2 ; } v1v1 =S. V v1 Vv2+V. Vv2Vv1 = Sv1v2-V v1v2⋅

Kv12 = Kv2Kv, = (— v2) (— v1) = v2v1 ·
V2V1

Κυυ,

67°. Suppose the plane of v, to be at right angles to the plane of v2, and the direction of rotation to be such that

Further, the versors of v1v, and v2v, (k and k) are perpendicular to the versors of both v, and v2 (i and j); or, the plane of v12 (and consequently the plane of its opposite, vv1, or v1v2) is perpendicular to the plane of v, and to the plane of v2.

Hence, the product of any two right quaternions in rectangular planes is a third right quaternion (vv) in a plane rectangular to both, which is changed to its own opposite (vv2) by reversing the order of the factors (Hamilton). In symbols,

√v1v2 = Vv ̧Vv2.

SECTION 4

On Circular Vector-Arcs

68°. Let O be the centre of a sphere of unit-radius. Then any arc AB of any great circle of the sphere may be regarded as the representative of the versor

OB

OA

For the

plane of the versor is the plane of the arc; the angle of the versor is measured by the length of the arc; and the direction of rotation is indicated by the direction in which the arc AB is drawn from A to B, fig. 24.

Definition. Two vector-arcs are equal, and only equal, when the origin and term of the first can be brought to coincide simultaneously with the origin and term of the second, by sliding the first backwards or forwards on its own great circle.

Thus, if on sliding (either way) the

arc AB round the great circle of a unit-sphere, shown in fig. 24, the point B coincides with D when A coincides with C; then,

AB = CD.

Two consequences follow from this definition. First, no two vector-arcs of the same great circle are equal, unless the direction of both, as seen from either pole of the common great circle, is towards the same hand. Secondly, whatever their length, no two vector-arcs of different great circles can be equal, except in one particular This case occurs when both the arcs are great semi

case.

A
FIG. 24.

circles.

All great semicircles are equal vector-arcs, since they

all represent versors of the form

of 1 is indeterminate.

ОА

=

1, and the plane

ОА

69°. Let any two arcs, C'C and AA', of different great circles bisect each other in B, fig. 25. Join A and C, C' and A', by arcs of great circles, and let the versors of any two quaternions, reduced to the form

A

The multiplication of versors is thus reduced to the addition of circular vector arcs.

70°. Unlike the addition of rectilinear vectors, and of quaternions, the addition of diplanar vector-arcs is not commutative;

BC + AB AB + BC.

;

For BC+AB=AC ; and AB + BC = BA' + C'B=C'A'. But ACC'A', although the two arcs are of equal length. For, if CC and ÂÀ' are both less than great semicircles (as

shown in fig. 25), or if one of the two is a semicircle and the other less than a semicircle, in both cases AC and C'A' belong to two distinct great circles, and are therefore unequal by definition. Were CC and AA' both semicircles, AC and

C'A' would both belong to the same great circle, of which B would be a pole; but they would have contrary directions, and would therefore be unequal by definition. In every case, therefore, ACCA', and, consequently,

AB + BC BC + AB ;

or, the addition of diplanar vector-arcs is not commutative.

72°. The addition of coplanar vector-arcs, however, is commutative; for, evidently, fig. 25,

BC+

BC + C'B= C'C = C'B + BC..

(1)

These equations show that the multiplication of coplanar quaternions is commutative, since they are equivalent to

73°. For the same reason that AB represents the versor

Hence, if a vector-arc represents the versor of any quaternion, the revector-arc (or the arc reversed) represents the versor of the reciprocal, or of the conjugate, of the quaternion. Consequently, fig. 25,

74°. If C'C and AA', fig. 25, are both great semicircles, AB (= Uq) and BC (= Uq') will be quadrants, i.e. q and q' will be right quaternions; and C'A' and AC will belong to the same great circle, but will have contrary directions. Therefore, since C'A' and AC are equally long, CA' is the revector of AC; and since AC = Uq'q,

Equation (1) is simply equation (3) of 66°, in different

symbols,