68] ON THE APPLICATION OP QUATERNIONS TO THE THEORY OF ROTATION. 407 
or since (if Aj be the quaternion for the combined rotation) lb = AjIfAj -1 , we have 
clearly 
Ai = M 1 A'A, 
M 1 denoting the reciprocal of the real part of A'A, so that 
Mr 1 — 1 — AA' — ¡jbfjb' — vv. 
Retaining this value, the coefficients of the combined rotation are given by 
Aj = M x (A + A/ + /iv — fiv') , 
fa = M x (/a + /x' + v'X — vX') , 
v x = M 1 (y + v' + X'y — X/jl); 
to which may be joined [if fa = 1 + Aj 2 + fa* + v*], 
fa = Mi kk, 
k, k, fa as before. A or A' may be determined with equal facility in terms of 
A', A 1? or A, A 1 . These formulae are given in my paper on the rotation of a solid 
body (Cambridge Mathematical Journal, vol. III. p. 226, [6]). 
If the axis L' be fixed in the body and moveable with it, its position after the 
first rotation is obtained from the formula Eb = All A -1 by writing 11= A' — 1. Repre 
senting by A" — 1 the corresponding value of lb, we have A" = AAAu 1 , which is the 
value to be used instead of A' in the preceding formula for the combined rotation, 
thus the quaternion of rotation is proportional to AA'A -1 A, that is to AA'. Hence 
here 
A x = Mi AA', 
which only differs from the preceding in the order of the quaternion factors. If the 
fixed and moveable axes be mixed together in any order whatever, the fixed axes 
taken in order being L, L',... and the moveable axes taken in order being L 0 , L 0 '... 
then the combined effect of the rotations is given by 
Ai = M ... A"A'AA 0 A 0 '... , 
M being the reciprocal of the real term of the product of all the quaternions. 
Suppose next the axes do not pass through the same point. If a, S, 7 be the 
coordinates of a point in L, and 
r = ai + @j + yk, 
then the formula for the rotation is 
n, — T = A (n — T) A“ 1 , 
or IIj = AnA -1 - (ArA -1 - T), 
where the first term indicates a rotation round a parallel axis through the origin, and 
the second term a translation.