16 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



spheres only are to be considered, the resulting formula?, 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. Cel. 

 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 F, but also admit of various other 

 applications of dcr. 



Suppose, therefore, J^''' is a function of 9 and trr, of the form con- 

 sidered by Laplace {Mec. Cel. Liv. iii.), r, 9, -zs- being the polar co-ordi- 

 nates referred to the axes JT, Y, Z, fixed in space, so that 



ar = r cos 0, y = r %\w9 cos Tsr, x = /• sin sin vr ; 



then, if we conceive three other fixed axes Xi, Y^, Z,, having the same 

 origin but different directions, P'^'^ will become a function of 0, and •zjti, 

 and may therefore be expanded in a series of the form 



(6) r <^> = r/"' + F.*'> + F/^' + F/^' + &c. . 



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 



^ - = f r' - 2rr' [cos 9 cos ff + sin 9 sin & cos {tn- - •sr')] + r'^\ "* 



as has been shewn by Laplace in the third book of the Mec. Cel., where 

 the nature of the different functions here employed is completely ex- 

 plained. 



In like manner, if the same quantity is expressed in the polar co- 

 ordinates belonging to the new system of axes X-,, F„ Z,, we have, 

 5ince the quantities r and r' are evidently the same for both systems, 



{^p, p) r \^ r r IT I 



^nd it is also evident from the form of the radical quantity of which 



