548 Prof. Orr on the Forced Precession and Nutations of 



a be the mean value of 6 and yfr the mean value of yjr, 

 equation (10) is thus equivalent to 



<f>= (n — yfr cos a) t. 



Equations (11) (12), on substituting the values of o) l5 co> 2 

 given by (9), thus lead to 



^sin^=M^+ M -^co8«)yj n (Q»__ A //)_^A // }. 



If we now write k=o-—n, cr will denote the speed of the dis- 

 turbing couple relative to axes in the plane of the equator 

 which rotate relative to the shell with a velocity equal and 

 opposite to that of the shell, which axes move very slowly in 

 space. The values just obtained may then be written 



d = «M^-*o«»«)7( ?l C ,, -o-A // ), .... (13) 

 ^sin^=M6^-^ cosa) V(^C""-crA // ) (14) 



The speed of the nutation relative to the nearly fixed axes 

 is thus not cr but cr — yjr cos a. . Inserting the values of A", 

 C /f the denominator here may be written in either of the 

 forms 



. n{C + c/)-*(A + A')-(A>-^)?^^ . . (15) 



or 



nKj — (rA + C'-^^ v . — - — : — L , . . . (16) 



(l-t-e)(<7 + en) v ; 



If the shell and contents formed one rigid body this 

 denominator would be replaced by ?z(C + C) — cr(A + A ; ). 

 We will suppose that C is greater than A, which is the case 

 if the outer surface, like the inner, is oblate. It thus appears 

 that if a be positive (and cr be less than n, which is true in the 

 particular cases to be considered) the disturbance of the shell 

 is always of the same sign as if the whole were rigid but 

 greater in amount. If cr be negative, then, when cr is 

 numerically less than en, the disturbance is of the same sign 

 as if the whole were rigid, but less in amount ; when cr is 

 numerically equal to en the disturbance is zero ; w r hen a is 

 numerically greater than en the sign of the disturbance 

 depends on the thickness of the shell ; when the thickness is 

 zero the disturbance is of sign opposite to what it would be if 

 the whole were rigid, and of amount which may be either greater 

 or less ; the magnitude increases with the thickness of the shell 



