Application of Quaternions to the Theory of Rotation. 197 



Suppose, in the first place, II = A— l,then IT 1 = A — 1, which 

 evidently implies that the point is on the axis of rotation. The 

 equation 11! = II gives the identical equations 



A(a — l)-fju,/3 + vy=0 



Aa' + ju,(/3'— 1) + ></ = () 



Aa" + ft/3" + vy" = 0. 

 From which, by changing the signs of A, jw,, v, we derive 



A(a — l)-ff*a' + va" = 



A/3+^(/3'-l)+v/3" = 



x 7 . fjX y +v ( 7 "-i)=o. 



Whence evidently, whatever be the value of II, 



AUA~ l -U=O t 



if after the multiplication if J, k are changed into A, ^, v, a 

 property which will be required in the sequel. 

 By changing the signs of A, jw>> v, we also deduce 



A -i nA=z'.(a# + a'j/ + a"#) 

 +i03# + /3'3/+/3"*) 

 + ^.(7^ + 7^ + 7"^), 



where a, /3, 7, «', /3', y', a", |3", y" are the same as before. 



Let the question be proposed to compound two rotations 

 (both axes of rotation being supposed to pass through the 

 origin). Let L be the first axis, A the quaternion of rotation, 

 L' the second axis, which is supposed to be fixed in space, so 

 as not to alter its direction by reason of the first rotation, 

 A' the corresponding quaternion of rotation. The combined 

 effect is given at once by 



n^A'CAIIA-^A'- 1 , i.e. 



II^A'A.II.tA'A)- 1 . 

 Or since if A t be the quaternion for the combined rotations 

 n 1 =A 1 nA7 1 , we have clearly 



A^M^'A, 

 M l denoting the reciprocal of the real part of A' A, so that 



Mf ■ = 1 - AA' - /V — vv'. 



Retaining this value, the coefficients of the combined rotations 

 are given by 



Ai = Mi ( A + A' + /x/v - /*v') 



jw^M^ + ^' + v'A-vA') 



v^M^v-K + AV-Ai*'); 



