546 Prof. Orr on the Forced Precession and Nutations of 



the shell alone were acted on by impulsive forces which gave 

 the shell additional velocities il 1? H 2 , fl 3 ; let o) l5 co 2 , g> 3 be the 

 actual velocities of the shell, m the mass of the fluid, a, a, y 

 the semiaxes of the cavity, A, A, C the principal moments of 

 inertia of the shell, A', A', C those of the fluid ; let 



ti=lm(*>-p, ^ 6=(« 2 -^)/(^ + 7 2 )- 

 The disturbing couple is of course supposed to have its axis 

 in the plane of the equator ; in the first instance let it be 

 constant in magnitude, and let its axis turn relative to the 

 shell with constant angular velocity k in the same direction 

 as the shell rotates. The component couples about the axes of 

 x, y may then be written as the real parts of Le*' , Me**' where 

 L = Wi. 



We have the kinematical equations 



©!=! + !!! ^ 



G) 2 =77 + X1 2 > ; (1) 



«3 = f+X2 3 ) 

 the dynamical equations 



(A + A / )«i-(A / -)U€)0 1 ~-|(A + A / )a) 2 -(A / -/ie)n 2 }a; 2 

 (A + A / )fi>«-(A / -/Ae)Xi s + {( k A + A> 1 -(A , -^€)n 1 }G) a 



(2) 



-(Co> 3 + C'f)G>i=M 



m 



00,3 + 0^=0; (3) 



and the equations of vortex motion 



• , „ f.du du v du~] 



97— J®! + £ fi) 3 =efgl 1} 



^-^2 + ^1 = ^^2—7]^) (5) 



If now the system be supposed never to deviate far from a 

 condition of rotation like a rigid body round the polar axis, 

 the quantities £l lf Xl 2 , Xl 3 , co v co 2 , f, tj are all small ; and 

 neglecting the products of such small quantities, from the 

 equations (3), (5), we deduce 



a) 3 = constant, ?= constant ; 



these constants we suppose of course to be equal ; let each be 

 denoted bv n. 



