determining the Thickness of the Earth's Crust. 101 



putting 



Cp'd.a'*=f(a), 

 Jo 

 we have the value of L for fluid pressure 



by integration and a property of Laplace's functions 



= 2a 3 ef(a) -3 • — fi\/l —y? sin <£= 3 / (fl)cosflsinflsin<ft. 



Similarly the value of the moment M for fluid pressure is 

 = ^-3— j[a) cos sin o cos <p. 



There will be similar terms for the moon. 



6. Hence the final alues for the moments L and M are 



( n a , 87 ™ 3e t< n Y 3 ^ a ■ a ■ jl , 3 ^ zi • a • _i "\ 

 f C—A-\ — Tr—j{ a ) )( — g" cos " sin "sin 9 H y-cos^sin^sin^ J 



( n A , 87ra 3 e„ N Y 3 ^ ... ,. 3M a . n , \ 

 — [ C— A-\- f(a) )( -g-cosc/smflcosft) -f — g-cose^sint^cosipj. 



Put 



then the differential equations become 



A ~+(C—A)n(Oci={Q —A) (—2- cos OsinO sin(/> 



, 3M' /) • zj ■ *\ 

 H 3- cos u l sm 0, sin 9, J, 



A d ^ ~(C-^)wa) 1 = -(C'- A) (^ 'cos sin 6 •cos <£ 



-3- cos f sm 0, cos 9, J. 



These equations are precisely the same as the ordinary equations 

 for finding precession and nutation in a solid body, S ! and M' 

 being put for S and M. The solution, therefore, putting P for 

 the precession of the crust, leads to 



* Mechanical Philosophy, p. 437- 



SM' 



