to the Theory of Rotation. 199 



ifjt k are changed into X', p', v', which proves that the trans- 

 lation is in a plane perpendicular to the axes. 



Any motion of a solid body being represented by a rotation 

 and a translation, it may be required to resolve this into two 

 rotations. We have 



n l =A,nA r i + T, 



where T is a given quaternion whose constant term vanishes. 

 Whence, comparing this with the general formula just given 

 for the combination of two rotations, 



A,— A'A.-M, 



T^-CA'r'A'- 1 — r')— A'CArA- 1 — r)A'-», 



the second of which equations may be simplified by putting 

 A'-'TA'=S, by which it may be reduced to 



s=(A'- l r'A'-r')-(ArA- 1 -r) > 



which, with the preceding equation A 1 =M I A'A, contains the 

 solution of the problem. Thus if A or A' be given, the other 

 is immediately known; hence also S is known. If in the last 

 equation, after the multiplication is completely effected, we 

 change i 9 J, k into A, ]«., v, or A', (*', v', we have respectively, 



s=A'- l r'A'-r', Szr-CArA-'-r), 



which are equations which must be satisfied by the coefficients 

 of JT" and r respectively. Thus if the direction of one axis is 

 given, that of the other is known, and the axes must lie in 

 certain known planes. If the position of one of the axes in 

 its plane be assumed, the equation containing S divides itself 

 into three others (equivalent to two independent equations) 

 for the determination of the position in its plane of the other 

 axis. If the axes are parallel, A, [x,, v are proportional to X 1 , jw/, v' ; 

 or changing i t j % k into X, p, v, or A.', ju/, v', we have S = 0; or 

 what is the same thing, T = 0, which shows that the transla- 

 tion must be perpendicular to the plane of the two axes. 



W Pi q> f have their ordinary signification in the theory of 

 rotation, then from the values in the paper in the Cambridge 

 Mathematical Journal already quoted, 



/ • • r v ^ dA - t dx. 



but I have not ascertained whether this formula leads to any 

 results of importance. It may, however, be made use of to 

 deduce the following property of quaternions, viz. if A^MjA'A, 

 M! as before, then 



1 / </A, dK,\ 1/ <ZA ■ dx.\ 



in which the coefficients of A' are considered constant. 



