ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 397 



of w, the function V has the property of satisfying a partial differen- 

 tial equation of the second order, and by an application of the Cal- 

 culus 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 unchanged, 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 endeavoured to ex- 

 plain 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 rela- 

 tions, the rational and entire factor in p still remains rational and 

 entire of the same degree, and may vmder its altered form be ex- 

 panded 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 h, whose differential coefficient is an algebraic quantity. 

 Moreover the first is immediately deducible from the corresponding 

 part of p 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 observation, and the discontinuity before 

 noticed falls upon one of the independent variables, in functions of 

 which both these attractions are expressed ; this variable being con- 

 stantly equal to zero so long as the attracted point j) remains within 

 the ellipsoid, but becoming equal to a determinate function of the co- 



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