and 



But 

 so that 



Hence 

 or 



RIGID BODY ABOUT A FIXED POINT. 289 



£ = x x y 



xx ] " = « 



XX l a + XX 1 « = o. 



t- - X~ X XX~ X 7 

 = -Y.tix,-i 7 = Y& = V. ^-^ 



^ »j = — V ij^jj . 



These are the equations we obtained before. Having found n from the last 

 we have to find x from the condition 



X~ X X a = V>! a • 



40. We might, however, have eliminated *i so as to obtain an equation con- 

 taining x alone, and corresponding to that of § 21. For this purpose we have 



n = <p-i I = <p- l x~ l y , 



so that, finally, 



X x X a = V . <p~ l x x 7 a 



or 



X l * = Y . x -i a<p~ x %- x y , 



which may easily be formed from the preceding equation by putting x -1 « for a. 



and attending to the value of x _1 given in last section. 



41. We have given this process, though really a disguised form of that in 

 §§ 19, 21, and though the final equations to which it leads are not quite so easily 

 attacked in the way of integration as those there arrived at, mainly to show how 

 free a use we can make of symbolic functional operators in quaternions without 

 risk of error. It would be very interesting, however, to have the problem worked 

 out afresh from this point of view by the help of the old analytical methods : as 

 several new forms of long-known equations, and some useful transformations, 

 would certainly be obtained. 



42. As a verification, let us now try to pass from the final equation, in x 

 alone, of § 40 to that of § 21 in q alone. 



We have, obviously, 



ar = q aq- 1 = x a 



which gives the relation between q and x 



VOL. XXV. PART II. 



a 1 



4e 



