Fry — The Centre of Gravity^ S^'c. 



Multiply the differential equation 



'n {n + 1) ffi' dp^ 



141 



d^ 

 da- 



^h- 



«.■' xp da] 



by «"+', integrate from 5 to a, and reduce, getting 



^/t - {n + 1) a'^^/t + pda^'^li - ft'*+3|o/t - &«(^V - ^^ + l^^o) /o«'f^a = 0. 



Now divide by (X^"+-, integrate from 5 to a, and multiply by 2% + 1, getting 



2ti+l,, 1 



P<^ — ; + 



A h"(hho--oi+lh,) 



pa^da - 





|oa'c?a = 0. (2) 



Now the condition (1) must be slightly modified; for in it | pda"'-^^ is 

 equal to 



pda^'^'r, + h''*'po (h - ^o) + [ pda''^%, 



where rj is the coefficient for the surfaces of equal density in the earth 

 corresponding to the coefficient h, and rjo is its value at the surface of the 

 earth. Since then any solution satisfies (2), and since (1) is the condition 

 which must be satisfied, the necessary and sufficient condition to be satisfied 

 by the arbitrary constants in the general solution for h is obtained by adding 

 (1) to (2), and multiplying by ft^"+^, getting 



a , „ 47r 



+ &" (bhfl -n ->r 1 Jiq) pa'^da - y^^ (&/t/ + nh^ pa^da = 0, 



or of the form K + ^V"+^ = where K and K' are independent of a. 



As this condition must be satisfied by all values of a from & to a, -ff" and 

 K' must both vanish, and so the conditions to be satisfied by the arbitrary 

 constants in h are 



■6 rh 



(p - p^^da'^^f) + &"'"po/^ + h"{l)K' - n ^- lh)\ pa-da = 



Jo 



A {2n + 1) k hhj+nh, '''> 



pd ^-„ + 



J^m 



(3) 



pa^da 



[20*] 



