222 FIGURE OF THE EARTH. [ll 



Just as we formerly divided by K*, /c 2 2 , ... and took the difference of two 

 successive equations, so here divide the second equation by /c 2 , and then 

 differentiate it with respect to K : 



d (M \ r E K t TT f ' d . , . , 4 IT d . 5 , 4 de 



7- (-re +5 - + 4- p -j-(K 5 e)cta--r-g/3 -7- (/c 5 e) + -ir/3 - r = 0; 



dK \ K 3 / K 6 K 6 J d* 5 /C 5 r dK V 5 ^ C?K 



cZ , 5 N de 

 or since ^ (* e) = -7- + 5/c e, 



d (M \ , , E, TT f d 6 e 



-- 5 -f + 4 -j P T- (K 5 e) dK - 47T/D - = 0, 



/C 6 K 6 J Ko r rf/C V r /C 



the equation becomes 

 I'M 

 \ * ; 



or ~jT j 3 j- e + 5 - ,! 



remembering that -7 = 



Multiply by K 6 and differentiate again ; we shall thus remove the remaining 

 definite integral and obtain 



d*e 8irpK~ de / 87773* 6\ 



die 4 ~M d K + \ M ~ K*) * 



This will determine e, when p (and therefore M) is known in terms of *. 

 Its solution will introduce two arbitrary constants, but we must remember 

 that this equation has been derived by two differentiations from the actual 

 equation which we have to satisfy, and the two constants must be adapted 

 so as to satisfy this ; they will evidently depend upon JE a and <u 2 . The 

 equation may be put under a simpler form ; for 



47r/3* 2 _ 1 dM 



therefore the equation may be written 



,d 2 e 



or since j^- 877/3* + 47r* 2 



(M)-(~-^ + ~ 



\ M. UK K 



We cannot treat this equation further unless we know p in terms of * ; 

 but it is worth while to shew how it will solve the inverse problem, viz. : 

 Given the law of ellipticity, to find what the law of density must be. 



Return to the earlier form of the equation and replace 877/3* by 



K ; 



