276 CELESTIAL MECHANICS. l.VU. 31. 



conditions of this fluxional equation ai^e fulfilled by putting 



5 = ^ sin {pt-\->,)——-fM gin (nt-\-y) or =r ^ sin (pf+^) — 

 Cp 



A B 



— q, and u=:^ cos(jof + A) -^ j^ fji! cos (nt + y), or = C 



ID 



cos(2>f4-A) — p^r [, since d« becomes —€ cos (2?^ + A)^d* 



A B 



— _.dg, and/?2/d^=:^cos(/)^ + A)pd^— — - rdf : but since 



dg= 5Z?rpd<(360), A dj=z.:?^^' rdt, and d*— pMd/= 

 / TT") ^df=:rdf, and dwzi — ^ sin {pt+><)pdt— 



JO yl ^ C A 



■— dr,ps6t — € sm{pt+x)pdt — -p^qdt, and — - dr— — - — 

 Cp C Lp L> 



qdty whence du-\^psdtz=: — qdt"]. 



In this manner the problem is completely resolved, since 

 the values of s and u afford us 9 and (p in terms of the 

 time, and since -v^ is deduced from (p and t. If the quan- 

 tity ^ — 0, the plane of re' and y becomes the invariable 

 plane, to which the angles 9, ^, and %J/ have been referred 

 in the preceding section (355). 



§ 31. Of the motion of a solid body round a fixed axis. 

 Determination of the simple pendulum oscillating in the 

 same time with the hody. P. 88. 



36 1 . T H E o R E M . The vibrations of a gravi- 

 tating body, whatever may be its form, are 

 synchronous with those of a simple pendukim 



of the length -r? C being the rotatory inertia 



with respect to the axis of motion, or S {y"^-^ 



