THE EQUILIBRIUM OP FLUIDS. 129 



the second to a similar function of x, y, z and the rectangular 

 co-ordinates of p. 



To prove this, we may remark that the corresponding value 

 V will become 



F=//WJ6W sin & (1 - /y/(V, y', z'} f ; 



the integral being conceived to comprehend the whole volume of 

 the sphere. 



Let now the function /be divided into two parts, so that 

 /(*', y' , z'} =/ (*', y', z') +/ (*', y', z) ; 



/ containing all the terms of the function/, in which the sum ot 

 the exponents of x', y' y z is an odd number ; and/ the remaining 

 terms, or those where the same sum is an even number. In this 

 way we get 



F=F 1+ F 2 ; 



the functions F t and F 2 corresponding to/ and/, being 

 F x =fr 2 dr'd0'dvr' sin (I - r") Vi <X> #'> *') ^ 

 F 2 -fMdffdJ sin tf (1 - /*) V, (a/, /, O /- 



We will in the first place endeavour to determine the value F t ; 

 and for this purpose, by writing for x, y , z f their values before 

 given in /, 0', OT', we get 



the coefficients of the various powers of r' z in i|r (r 2 ) being evi- 

 dently rational and entire functions of cos ff, sin & cos OT', and 

 sin0' sintn-. Thus 



' sin (1 - r' 2 )? / ^ (r /2 ) ^- ; 



this integral, like the foregoing, comprehending the whole 

 volume of the sphere. 



Now as the density corresponding to the function V l is 



it is clear that it may be expanded in an ascending series of the 

 entire powers of a?', y, z, and the various products of these 

 powers consequently, as was before remarked (Art. 1), F, ad- 



9 



