134 THE LAWS OF 



3. The object of the preceding sketch has not been to point 

 out the most convenient way of finding the value of the function 

 V, but merely to make known the spirit of the method ; and to 

 show on what its success depends. Moreover, when presented 

 in this simple form, it has the advantage of being, with a very 

 slight modification, as applicable to any ellipsoid whatever as to 

 the sphere itself. But when spheres only are to be considered, 

 the resulting formulae, as we shall afterwards show, will be much 

 more simple if we expand the density p in a series of functions 

 similar to those used by Laplace (Mec. Gel. Liv. iii.) : it will 

 however be advantageous previously to demonstrate a general 

 property of functions of this kind, which will not only serve to 

 simplify the determination of V, but also admit of various other 

 applications of do; 



Suppose, therefore, Y (i) is a function of and r, of the form 

 considered by Laplace (Mec. Gel. Liv. iii.), r, 6, v? being the 

 polar co-ordinates referred to the axes X, Y", Z t fixed in space, 

 so that 



x = r cos 0, y r sin 6 cos -or, z = r sin 6 sin cr ; 



then, if we conceive three other fixed axes JT 15 Y t , Z 11 having 

 the same origin but different directions, Y (i) will become a func- 

 tion of # x and -zzTj, and may therefore be expanded in a series of 

 the form 



Suppose now we take any other point p and mark its various 

 co-ordinates with an accent, in order to distinguish them from 

 those of p\ then, if we designate the distance pp by (p, p'), we 

 shall have 



L.^ = [r 2 - 2rr (cos cos & + sin sin 0' cos (w -*')} + r'T* 



o) + Qd^ + ^ "I* + 0) r _l + &c \ 

 r r r ) 



as has been shown by Laplace in the third book of the Mec. Gel, 

 where the nature of the different functions here employed is 

 completely explained. 



