198 Mr. A. Cay ley on the Application of Quaternions 



to which may be joined 



x^M^.xx', 



x, x', Xj as before. Aj or A' may be determined with equal faci- 

 lity in terms of A', A M or A, A 1# These formulae are given in my 

 paper on the rotation of a solid body (Camb. Math. Journal, 

 vol. iii. p. 226). 



If the axis L' be fixed in the body and moveable with it, 

 its position after the first rotation is obtained from the formula 

 n 1 = x'VlIA- 1 by writing 11= A' — 1. Representing by A"— 1 

 the corresponding value of ITj, we have A"=AA'A -1 , which is 

 the value to be used instead of A' in the preceding formula for 

 the combined rotations, i. e. the quaternion of rotation is pro- 

 portional to AA'A _1 .A, that is to AA'. So that here 



A,=M,AA', 



which only differs from the preceding in the order of the qua- 

 ternion 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 , L' . . then the combined effect of the rotation is given by 



A, = M..A"A'AA A'o.., 



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 «, §, y be the co-ordinates of a point in L, and 



r = ai + gj+ yk, 



the formula for the rotation is 



n,-r=A(n-r)A- 1 , 



or 



n^AiiA^.-CArA-J-r), 



where the first term indicates a rotation round a parallel axis 

 through the origin, and the second term a translation. 

 For two axes L, L' fixed in space, 



n^A'A.n.CA'A)" 1 . 



— (aT'a'- 1 — r')-A'(ArA- 1 -r)A'- 1 ; 



and so on for any number, the last terms being always a trans- 

 lation. If the two axes are parallel, and the rotations equal 

 and opposite, 



A = A'-»., 

 whence 



n^n + A'cr-r^A'-'cr-r') . 

 or there is only a translation. The constant term vanishes if 



