136 THE LAWS OF 



YJ h} , except when h i. Hence, the expansion (6) reduces 

 itself to a single term, and becomes 



We thus see that the function Y (i} continues of the same form 

 even when referred to any other system of axes X v Y v Z v having 

 the same origin with the first. 



This being established, let us conceive a spherical surface 

 whose center is at the origin of the co-ordinates and radius /, 

 covered with fluid, of which the density p = Y' (i} ; then, if da' 

 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 ty the characteristic of any function 

 whatever. I say, the resulting value of V will be of the form 



R being a function of r, the distance Op only and Y (i} what 

 F' (i) becomes by changing 6 ', /, the polar co-ordinates, into 

 0, OT, the like co-ordinates of the point p. 



To justify this assertion, let there be taken three new axes 

 X lt Y v Zv so that the point p may be upon the axis X^ ; then, 

 the new polar co-ordinates of da' may be written r', 0', -or', those 

 of ^> being r, 0, -cr, and consequently, the distance will become 



g = ^/ (r' 2 2**r' cos 0j' + r 2 ) j 



and as da r'^dO^d^^ sin #/, we immediately obtain 

 F= / F m r r "^ 1 cfer 1 sin 0^ (r 2 - 2rr f cos 0/+ r' 2 ) 



0j' sin 6^ ty (r z %rr' cos 



Let us here consider more particularly the nature of the 

 integral 



i 







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



