30 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



then the expression immediately preceding may be written 



■^J7\r 4<-n.6-n Q,t-2t' + 2 



(n-2 \ 2.4 2t-2t' 



dn [-^ .) 



sin 



and by giving to t the various values 0, 1, 2, 3, &c. of which it is sus- 

 ceptible, and taking the sum of all the resulting values of F/'' the quantity 

 thus obtained will be equal to V^^ or that part of V which is of the 

 form Y^\ Thus we get 



27r^ r' 



■fr(i), 



sm 



.«/)(0)./„« 





+ &c &c &c. 



since aU the terms in the preceding value of Vi-^ in which t'>t vanish 

 of themselves in consequence of the factor 



/ 2^-2^ +4-w \ 



= (when t > t). 



2 . 4 2i?-2^ „,. ., . _ „ /4-w 



(-'■«) rC-?) 



But F"^'^ as deduced from the given value of V may be expanded in 

 a series of the form 



r«=?''. {r„®-i- r;(v^+ r,»./^+ v^'>t^+kc.\ 



and if in order to simplify the remaining operations, we make generally 



__„ 27r^ n-2.n M+2/-4 n-\.n + l n + 2i+2t-3 ^„., 



' . [n-2 \ 2.4 2t 3.5 2« + 2^+l ' 



sm [—.] 



