18 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



of which the density p = P''*'' ; then, if d<r' represent any element of 

 this surface, and we afterwards form the quantity 



the integral extending over the whole spherical surface, g being the 

 distance p, da and y\f the characteristic of any function whatever. I 

 say, the resulting value of V will be of the form 



V= Y^B; 



R being a function of r, the distance Op only and K<'' what Y'^^ becomes 

 by changing 9', w, the polar co-ordinates, into 9, tit, the like co-ordinates 

 of the point p. 



To justify this assertion, let there be taken three new axes JT,, I^„ Z„ 

 so that the point p may be upon the axis Xx ; then, the new polar 

 co-ordinates of da' may be written r', ff, tjt', those of p being r, 0, •sr. 

 and consequently, the distance will become 



g = ^{r" - 2 rr' cos 9,' + r^) ; 



and as da^' = r'^d9i'd'sri sin 9,', we immediately obtain 



r = fY'^'Vde.d-sr, sin 9, f (/•-- 2rr' cos d,' + O 



= r'^SZd9; sin 0/ ^{r'-^rr' cos 0/ -f r'^)f^Zd-ur( Y' <". 



Let us here consider more particularly the nature of the integral 



In the preceding part of the present article, it has been shown that 

 the value of Y'^'^, when expressed in the new co-ordinates, will be of 

 the form P'/*'' ; but aU functions of this form (Vide Mec. Cel. Liv. iii.) 

 may be expanded in a finite series containing 2 « + 1 terms, of which 

 the first is independent of the angle "sr,', and each of the others has 

 for a factor a sine or cosine of some entire multiple of this same angle. 

 Hence, the integration relative to ro-/ will cause all the last mentioned 

 terms to vanish, and we shall only have to attend to the first here. 

 But this term is known to be of the form 



, / ,. i.i — \ ,. „ i.i—l.i-2.i — S ,,• , « N 



