Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 29 



f being the characteristic of a rational and entire function of the same 

 degree as V, and which we will here endeavour so to determine, that 

 the value of V thence resulting, may be equal to any given rational 

 and entire function of x, y, % of the degree s. 



Then by Laplace's simple method {Mec. Cel. Liv. iii. No. 16.) we 

 may always expand F" in a series of the form 



r= r<«> + r(» + r® + &c + r«. 



In like manner as has before been remarked, we shall have the 

 analogous expansion 



f{x',y', ,')=/''«' +/'<'>+/'^=>+/'<'>+ &c +/'«, 



of which any termy*'' for example, may be farther developed as follows, 



/'« =^'('V' +y;'''V"+'^ +//»/'+* + &c. = r" (/'« +y;'('V'^ +/'»;.'^ + &c.) 



y", yj'<'>, j^''*^, &c. being quantities independent of / and all of the form 

 K'"' {Mec. Cel. Liv. iii.) Moreover F/" the part of F' due to the general 



term Jl'^'^r''+^* of the last series, will be obtained by writing for (i 



in the equation (11), and afterwards substituting for 



(n — 2\ _ f4i-n 



r(!t^)r(l^) us value- 



n-2 

 sin 



In this way we get 



27r;/;'V ±-„,e-n 2t-2f + 2-n 



' . fn-2 \ 2.4 2t~2t' 



sm (-^.j 



n-2.n w + 2#'-4 w — 1 . w + 1 n + 2i + 2f — 3 ^ 



^ 2.4 2? ^ 3 . 5 ...... 2i + 2t' + l ' 



yj<'^ being what J]''-^ becomes by changing 6', -ar' into 6, sr, and the finite 

 integral being taken from t' = to t'=<x . 



Let us now for a moment assume 



^(0= 



n-2.n w + 2#' — 4 «-l.« + l n + 2i + 2t'-3 



X 



2.4 2t' 3.5 2i + 2t' + l ' 



