RIGID BODY ABOUT A FIXED POINT. 297 



These values may be called ± m 15 ± /* 2 , and we have 



M>1 + f> 2 — U ■ 



52. The complete solution of the equation (42) is therefore 



q = Q x cos [h-fi + Q 2 sin (h x t + Q 3 cos p 2 t + Q 4 sin p 2 t . 



This, however, is far too general for the solution of the original problem, for it 

 involves sixteen arbitrary constants instead of four. But it is a mere piece of 

 ordinary analysis to find twelve of these in terms of the other four. 

 Thus, let us write 



Q, = H x + Iji + J J + K^ , 

 Q 2 = H 2 + Ij* + J 2 i + K 2 & , 

 Q 3 = H 3 + I,* + J 3 / + KJc , 

 Q 4 = H 4 + IJ + J J + KJt . 



If these values be substituted in the above expression for q, and the resulting 

 value of q be used in the equation 



q = Vai -f h(J cos nt + k sin ntj] q , 



we find, on replacing products of sines and cosines of multiples of t by sums of 

 sines or cosines, two sets of terms. One of these is of the type 



cos (n — fi^t , 

 which, being equal to 



COS fl 2 t , 



may be allowed to remain in the equation. The other set is of the type 



COS (% + /C6j) t , 



and the terms introducing it must vanish identically. 



This consideration gives us the following relations among the sixteen con- 

 stants above 



I 2 = Hj , H 2 = — Ij , <I 2 = — Kj , K 2 = J j , 



I* = H 3 , H 4 = — I 3 , J 4 = — K 3 , K 4 = J 3 ; 



so that the values of eight are assigned in terms of the remainder. 

 Next, by equating coefficients of each such distinct term as 



i cos fhjt , k sin fi 2 t , &c, 



we obtain sixteen additional equations, of which, however, eight are mere repeti- 

 tions of the other eight. Rejecting them, we find the remainder to be 



VOL. XXV. PART II. 4 G 



