THE EQUILIBRIUM OF FLUIDS. 135 



In like manner, if the same quantity is expressed in the 

 polar co-ordinates belonging to the new system of axes 

 JTj, Y l , Z^ we have, since the quantities r and r are evidently 

 the same for both systems, 



and it is also evident from the form of the radical quantity of 

 which the series just given are expansions, that whatever number 

 i may represent, Q will be immediately deduced from Q (i} by 

 changing 0, w, ff, is into 1? VF 19 #/, w/. But since the quan- 



9* 



tity is indeterminate, and may be taken at will, we get, by 



equating the two values of . -- ^ and comparing the like powers 



v 



of the indeterminate quantity , 



<2 = <?,'. 



If now we multiply the equation (6) by the element of a 

 spherical surface whose radius is unity, and then by Q (h) = Q^ h \ 

 we shall have, by integrating and extending the integration over 

 the whole of this spherical surface, 



fdfjidv Q (h) Y^ =fd^d^ QM { YW + Y" + Yf> + &c.}. 



Which equation, by the known properties of the functions 

 Q (h} and Y (i \ reduces itself to 



when h and i represent different whole numbers. But by means 

 of a formula given by Laplace (Mec. CeL Liv. iii. No. 17) we 

 may immediately effect the integration here indicated, and there 

 will thus result 



4>7r y(h) . 

 ~2h + l** 



Y^ (h) being what Y^ becomes by changing 9 V ^ into #/, r t ', 

 and as the values of these last co-ordinates, which belong to p\ 

 may be taken arbitrarily like the first, we shall have generally 



