190 ON THE DETERMINATION OF THE ATTRACTIONS 



ing directly the value of V, all its differentials have first been 

 deduced, and thence the value of V obtained by integration. 

 This slight modification has been given to our method, both 

 because it renders the determination of V in the case considered 

 more easy, and may likewise be usefully employed in the more 

 general one before mentioned. The other application is remark- 

 able both on account of the simplicity of the results to which it 

 leads, and of their analogy with those obtained by Laplace. 

 (Mec. Cel. Liv. ill. Chap. 2). In fact, it would be easy to shew 

 that these last are only particular cases of the more general ones 

 contained in the article now under notice. 



The general solution of the partial differential equation of 

 the second order, deducible from the seventh and three following 

 articles of this paper, and in which the principal variable V is 

 a function of s + 1 independent variables, is capable of being 

 applied with advantage to various interesting physico-mathe- 

 matical enquiries. Indeed the law of the distribution of heat in 

 a body of ellipsoidal figure, and that of the motion of a non- 

 elastic fluid over a solid obstacle of similar form, may be thence 

 almost immediately deduced; but the length of our paper en- 

 tirely precludes any thing more than an allusion to these appli- 

 cations on the present occasion. 



1. The object of the present paper will be to exhibit certain 

 general analytical formulae, from which may be deduced as a 

 very particular case the values of the attractions exerted by 

 ellipsoids upon any exterior or interior point, supposing their 

 densities to be represented by functions of great generality. 



Let us therefore begin with considering p as a function of 

 the s independent variables 



and let us afterwards form the function 



' ' ' ' 



n 

 P 



the sign / serving to indicate s integrations relative to the vari- 

 ables a?/, a? 2 ', # 8 ', ... x t ' t and similar to the double and triple ones 



