On the Motion ofaFendulum affected by the Earth* s Rotation, 377 



But 



^=^cos^, 'v/r sine =s;)^ sin ^, 



retaining only infinitesimals of the first order. Whence 



p^2 = <^2 + i|r2 sin^ Sj 



■^ sine=tan^. 

 Consequently 



and therefore, 



ij?=r cos (€ + </))+ -^(^^4- ^/r^sin^e) cose, 



2/ = r sin (e + </>) sin (y5^ + '^) + -^ (0^ + 'v/^^ sin^ e) sin e sin ^t^ 



^-ssr sin (e + <^) cos [fit + '^) + o; (^^ + "^^ sin^ e) sin e cos fit. 



Substituting therefore in (1.), observing that the virtual velo-^ 

 cities S(f>j S\jr are geometrically independent, and neglecting 

 small quantities of orders higher than the first, we obtain 



^+Af+B^ + C=0, (3.] 



^^A'f +B'^ =0, (3.) 



in which 



A=-/8sin2e, B = |— /32cos2e, C=~|-sin3e, 

 A'=;Scose, B'=|^. 



Q 



Write H— ^ for <^, and equations (2.), (3.) become 



^+Af ^B'^=0. ...... (5.) 



These equations are linear and of the second order. We shall 

 obtain a particular solution of them by assuming 



n=-Re''\ yfr^nke"^'. 



Substituting in (4.) and (5.), we have for determining k, m, 



m2+A^m+B=0,. 



km^+Mm-i'B'k^O, 



