ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 425 

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



= Pe,e,e3 e,_,; 



P being a function of p only, and afterwards generally 6, a function 

 of dr only. In fact, if we substitute this value of (p in (40), and then 

 divide the result by ^, it is clear that it will be satisfied by the system 



e,,.,</e\_, 



— ^<- 1 



d'Q,.^ , _ cos0,_2 o?e,_2 , X._, ^ .^^^ 



Os-2 de^._s ' sin 0,_2 9s_2</0s_2 sin 6^,-2 



+ 2 . -7— -p^ — 77 j7i h -; — —^ — = X,- 



B._3d0\.3 ' sin 9^-3 Qs-3d9,_3 sin0^ 



&;c. &c. &c. &c. 



combined with the following equation, 



d'P s-l-np' dP \, k ^ 



P^p" ^ /" (1 -p') ■ P«?/' />' 1 -p' 



where k, X,, X^, X3, &c. are constant quantities. 



In order to resolve the system (41), let us here consider the general 

 type of the equations therein contained, viz, 



- ^'Q'- , (r-i\ ^"^^-- '^^^ + ( ^'-^' X "i ft 

 d9\_,. ^ >sm9,_/d9,.r \sm9\., a,_,j «,.,. 



Now if we reflect on the nature of the results obtained in a preceding 

 part of this paper, it will not be difficult to see that 6,_r is of the form 



e,_. = (sine._,)*;j = (1-M^)«;>; 



where j9 is a rational and entire fimction of m = cos0s_r, and / a whole 

 number. 



By substituting this value in the general type and making 



\..r^i = - i{i + r - 2) (43) 



we readily obtain 



= {1-M.')^: -{2i + r)^^- {X._. + i{i + r-l)}p. 



