Rigidity of the Earth. 



493 



We have now to express B in terms of A , A 2 , and s. 

 When this is done, we can express the coefficients A 4 , A 6 , &c. 

 in terms of two arbitrary constants A and A 2 , and these can 

 be found from the boundary conditions. 



Thus 



B 



\ 2m + 2) 



A, 



whence 



5 7 4 



= 2A + -A 2 + A 4 + gA 6 + -A 8 +... 



= 0-0765 B + . 1-751 A + 1-207 A 2 + 0-01172*, 



B = 1-896 A + 1-307 A 2 + 0-01269*. 



. . . (45) 



The substitution of this value for B in equations (37) and 

 (40) to (44) gives 



A 4 = 10- 4 {-1114A + 1013A 2 + 128'94, ^ 



A 6 = 10- 5 {704-5A -5480 A 2 }, 



A 8 = 10- 6 {2101A + 4244A 2 -244-3 4, 



A 10 = 10- 6 {-415A +781A 2 + 41-l 5 }, . (46) 



A 12 = l()- 7 {32A -1130A 2 -f- 10-4 5 }, 



A 14 = 10- 7 {31A + 25A 2 -3-44, 



A 16 = 10- 8 {-26A + 65A 2 + 2-44. 



We must now make use of equations (30) and (30 «) to 

 determine A and A 2 . 

 The substitution 



mm 



e = H,,- 

 a m 



in equation (30) will give, after the differentiations have 

 been performed 



[1 r m-l 



g m(m - 2)(m + 3) (m + 5) — 



-g^(» + l)(»H««-12)5]-0. . (47) 



when r = a. 



But 



since 

















P^ 



(-- 



7-5 



.2 \ 



3 see 



that, 



when r = a, 















ldp _ 



p dr ~ 



1 



2 r 5 



15 

 a 





6 



a 



