398 Mr green, ON THE DETERMINATION OF THE 



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

 of seeking 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 remarkable both on account of 

 the simplicity of the results to which it leads, and of their analogy 

 with those obtained by Laplace. (Mdc. C^. Liv. iii. 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 1^ is a function of # + 1 

 independent variables, is capable of being applied with advantage to 

 various interesting physico-mathematical 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 entirely precludes any thing more than an allusion to these ap- 

 plications 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 



»r J , x<i , x^ ••••• o/i, 



and let us afterwards form the function 



dxjdx^ dxj dxl . p .^. 



'{{x,-xiJ^{x,-xl)^^ ^(x.-xlJ^u'-S^ 



r=f- 



n-1 

 2 



