a Rotating Ellipsoidal Shell containing Liquid. 551 



: . $= f KI/ + M') 



111 (.n(0 + CO- t (A + AQ- (A/ ~^'("-'? 



€11 + S 



* (L/ " M V-#ieWn + ,) l C08rt - (19) 



fi(0 



If the shell be supposed without mass, and e, es/n small, 

 the denominators then becoming approximately C / €n 9 /(en±, s), 

 these values are approximately 



L'en + Ws 



G'en? 



sin st, (20) 



^sin0= — q7^2 — cosst (21) 



The corresponding values if the whole system were rigid, 

 which may be obtained from (18), (19) by neglecting C, A', p,, 

 and then changing C and A into C', would be approximately 



L'n + M's . 



# = - Q/^.^) Sln st > ' ' ' * * ' * ( 22 ) 



M'n + L's 

 -f sin0=Q/( w 2_ g s) cos^ (2d) 



If 1/ and M' be nearly equal (in the cases to be discussed 

 the several values of I//M' are approximately ^f, |, J), we 

 see that the nutation of the massless shell may be obtained 

 approximately from the value it would have if the whole were 

 rigid by multiplying by (1 + s/en) (1 — s/n). 



7. In the case of the nineteen-y early nutation the couples 

 about OA', OB', denoted above by — I/sins£, M'coss^, are 

 of the forms — K cos a sinpt, — K cos 2a cos pt, where p is the 

 mean speed of the nodes of the moon's orbit along the ecliptic ; 

 so that here s= —p and I//-M/ = cos a/ cos 2a. Replacing this 

 ratio by unity, we obtain Lord Kelvin's result that if e=g~Jg 

 and the shell be without mass its nutation would be about 

 |f times the value it would have if the whole were rigid. 

 More accurately, the semiaxes of the ellipse described by the 

 polar axis of a rigid earth being taken as 9 //k 22 perpen- 

 dicular to and Q f/ 'S6 parallel to the plane of the ecliptic, 

 those for the shell would be (9"-22 x -966 = ) 8"-91 perpen- 

 dicular to, and (6"-86 x'945 = ) 6" -48 parallel to the plane of 

 the ecliptic. 



If the mass of the shell be taken into account, a conside- 



