OF THE MOTIONS OF A SOLID BODY. 275 



principal axis. But if the value of n^ became negative, 

 and n were .consequently imaginary, the values of sin 

 {nt-\-y) and cos {nt-Vy) would be changed into exponential 

 or logarithmic quantities (358), and the expressions for 

 q and r might then increase indefinitely, and these quan- 

 tities would no longer be infinitely small, so that the mo- 

 tion would have no stability. Now the value of w is real if 

 C is the greatest or the smallest of the three quantities 

 Ay B, and C, for then the product {C—A) . (C—B) is 

 positive, but this product is negative when Cis of inter- 

 mediate magnitude, and n then becomes imaginary. 



360. Corollary. Retaining the same 



notation, if 9 be very small, we shall have 



A 



sm fl sin ^=^ sin (p^+^) — ^ /* sin (nt-^y), and 



sin dcos ^=^cos (pt-^^) — -^z^' cos {nt-^-y) ; ^ 

 and A being two new constant quantities. 



In order to determine the position of the axes with re- 

 gard to a quiescent space, we may suppose the third prin- 

 cipal axis very nearly perpendicular to the plane of x'' and 

 y'f so that we may be able to neglect the square of d, and 

 to make cos 5=1, we shall then find for the value o^pdt, 

 instead of d^ — d^/. cos 6, d(p — d%|/, whence 4^ = (p — pt — e, 

 s being a constant quantity. We have then, since qdt:=z 

 d^/. sin 9 sin p — d9. cos (p, and rd^ridx^. sin 6 cos (p-\-d9. 

 sin (p, putting sin 9 sin (pzzs, and sin 9 cos (p=u, dszzcos 9 

 sin (p d9 -{- sin 9 cos (pd<p=.sin (pd9+sin9 cos <pd(p, pudt=: 

 sin9 cos (p {d(p — d^^),ds—pudt—s\n<pd9-\-s\n9 cos ^d^^zzrd^; 

 and dwzzcos ^d5— sin 9 sin (pd^, psdtzz s\n 9 sin (p {d<p — d^^) 

 and du-\-psdt=co9(pd9 — sin d sin ^d>^= — qdt. Now the 



T 2 



