RIGID BODY ABOUT A FIXED POINT. 285 



Though remarkably simple, this formula, in the present state of the development 

 of quaternions, must be looked on as intractable, except in certain very particular 

 cases. 



31. Instead of solving the differential equation (7) of the group in § 21, having 

 previously eliminated n from it by means of the other two, we may solve the 



second equation of the group, 



71 = & (17), 



for q, and treat n as known in terms of £. £, of course, is to be regarded as found 

 by the processes of §§ 23, 25. As this mode of attack leads to a determination of 

 q by a set of three new differential equations, instead of the four of § 27, it may 

 be useful to consider it briefly, but only for the case of y — constant. Its interest 

 seems to be derived entirely from the quaternion investigation to which it leads. 



32. In consequence of (17), just cited, we may write 



q = 7* + K (25), 



which will be found to satisfy that equation, whatever value is assigned to S. 



But S is really not unrestricted in value ; for, if we exhibit it as the sum of 

 two vectors, thus 



of which S 2 satisfies the equation 



7h + ^ = , 

 or, which is the same thing, the pair 



S3 2 (7 + 0=0 



V3 2 (y-0 = 

 we see that 



satisfies both. [This depends on the fact that T£ = Ty]. Hence S must be de- 

 prived of its resolved part parallel to y — £ : or we must have 



s% -o = o (26). 



33. By differentiation of (25) we have 



i = 7* + K + *Z- 

 Substituting in (7) we have 



But, § 22, 

 whence 



and the above equation becomes 



2(y3 + bt) = yhn + hZ, (27), 



VOL. XXV. PART II. 4 D 



