OF ELLIPSOIDS OF VARIABLE DENSITIES. 217 



d^... d&^p"- 1 sin 6^ 2 sin 0^... s 



dp) p 2 l sin 0* sin 2 a ... sin 0V-i ' 

 and 1 Sf r 2 = 1 p 2 . 



Proceeding now in the manner before explained (No. 8), we 

 obtain for the equivalent of (39) by reduction 



d 2 (b s 1 np 2 dd) 

 = -^-T 4 



-p>) 'dp 



3 !'^. 

 *ff r d0 k 



a< P" 



But this equation may be satisfied by a function of the form 



P being a function of p only, and afterwards generally S r a 

 function of r only. In fact, if we substitute this value of <f> in 

 (40), and then divide the result by 0, it is clear that it will be 

 satisfied by the system 





, o CQS 0J-, ^-3 , X *- 2 _> 

 "T ""= ~7\ 7=\ 7?i " /12 /v 



&c. &c. &c. &c. 



combined with the following equation, 



d * P s-i-np* dP A, k_ 

 Pdp* + p(l-p*) 'Pdp + p* l-p*~ 



where k, \, X 2 , X 3 , &c. are constant quantities. 



