Mb GREEN, ON THE LAWS. OF THE -EQUILIBRIUM OF FLUIDS. 21 

 If now we can obtain the part of ^ due to the term 



we shall immediately have the value of V by summing all the parts 

 corresponding to the various values of which i and t are susceptible. 

 But from what has before been proved (Art. 3.), the part of V now 

 under consideration must necessarily be of the form F"*'^; representing, 

 therefore, this part by F"/'', we shall readily get 



r/" =/J/'+^'+^</r' (1 - ryjtl-nr'de' sin ^/'<'^^'-". 



Moreover from what has been shown in the same article, it is easy 

 to see that we have generally 



fV'^'^clu'de' sin e'^ig"-) = Stt F« ^f'f •:'^'~^ /-i'c?Mi' (i) yl^{r'-2rr',x,' + r") ; 



\(/ being the characteristic of any function whatever, and P'^'' what P''"* 

 becomes by substituting 9, w the polar co-ordinates of p in the place 

 of 6', TB-', the analogous co-ordinates of the element dv. If therefore 

 in the expression immediately preceding, we make 



F'«=/'« and fig^) =^'.- = (^^)^, 



and substitute the value of the integral thus obtained for its equal in 

 Vf-'^ there will arise 



where yj® is deduced from Jl'^^ by changing 9", %r' into 6, w, and (i), for 

 abridgement, is written in the place of the function 



,; i.i-1 ,;_; , i.i-l.i-2.i-3 ,^_^ . 

 '"~2.2e-l'" ^2.4.2i-1.2«-3'" *'''• 



As the integral relative to n\ which enters into the expression on 

 the right side of the equation (8) is a definite one, and depends therefore 

 on the two extreme valvies of fj.\ only, it is evident that in the deter- 

 mination of this integral, it is altogether useless to retain the accents 



