OF ELLIPSOIDS OF VARIABLE DENSITIES. 189 



be determined. But after the introduction of w, the function V 

 has the property of satisfying a partial differential equation of 

 the second order, and by an application of the Calculus of 

 Variations it will be proved in the sequel that the required 

 value of V may always be obtained by merely satisfying this 

 equation, and certain other simple conditions when p is equal to 

 the product of two factors, one of which may be any rational 

 and entire function of the s accented variables, the remaining 

 one being a simple algebraic function whose form continues un- 

 changed, whatever that of the first factor may be. 



The chief object of the present paper is to resolve the 

 problem in the more extended signification which we have en- 

 deavoured to explain in the preceding paragraph, and, as is by 

 no means unusual, the simplicity of the conclusions corresponds 

 with the generality of the method employed in obtaining them. 

 For when we introduce other variables connected with the 

 original ones -by the most simple relations, the rational and 

 entire factor in p still remains rational and entire of the same 

 degree, and may under its altered form be expanded in a series 

 of a finite number of similar quantities, to each of which there 

 corresponds a term in V, expressed by the product of two 

 factors ; the first being a rational and entire function of s of the 

 new variables entering into V, and the second a function of the 

 remaining new variable ^, whose differential coefficient is an 

 algebraic quantity. Moreover the first is immediately deducible 

 from the corresponding. part of,/)' without calculation. 



The solution of the problem in its extended signification 

 being thus completed, no difficulties can arise in applying it to 

 particular cases. We have therefore on the present occasion 

 given two applications only. In the first, which relates to the 

 attractions of ellipsoids, both the interior and exterior ones are 

 comprised in a common formula agreeably to a preceding obser- 

 vation, and the discontinuity before noticed falls upon one of the 

 independent variables, in functions of which both these attrac- 

 tions are expressed ; this variable being constantly equal to zero 

 so long as the attracted point p remains within the ellipsoid, but 

 becoming equal to a determinate function of the co-ordinates of 

 p, when p is situated in the exterior space. Instead too of seek- 



