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



= TTtt/'V COS i6 . —5 — . 



, „P n-1 .n + 1 n + 2f — 3 7i-l . ji + 1 n + 2i + 2t'-3 



" 2.4 2t' ^ 2 . i 2i + 2f 



3-n.5-n 1 + 2^-2^'-?? ^ 



^ 2(i + 5-n 2fi + 3 + 2t+2t'-n' 



the sign of integration 2 belonging to the variable f. 



Having thus the part of V due to the term a,''' cos i9' in the expansion 



of J'iaf, if) it is clear that we may thence deduce the part due to the 



analogous term J/'^ sin i& by simply changing «/" cos iQ into J/'' sin iO, and 



consequently we shall have the total value of V itself, by taking the 



sum of the various parts due to all the different terms which enter 



into the complete expansion of y(a;', y'). 



11 3 



If now we make iS = — - — and recollect that 



2 



sin 





the foregoing expression will undergo simplifications analogous to those 

 before noticed (Art. 5.) Thus we shall obtain 



TT^a/" , .^^„„M-1.« + 1 n-k2t-3 



r/" = "-:^ r' cos iQ . 2 r''' 



' . (n 

 sin I - 

 2 



n-1 \ 2.4 2t' 



sin — - — IT 



(- 



n-1 .n + 1 n + 2i + 2f-3 3-n. 5-n 1 + 2^-2^- w 



'^ 2 . 4 2i + 2t' ^2.4 2if-2^ 



or by writing for abridgment 



,. ^ n-1. n + 1 n + 2t'-3 n-l.n + 1 n + 2i + 2f-3 



"P^^'*^'- 2 . 4 2f "" 2 . 4 2i + 2t' ' 



there will result this particular value of /8 



,^(. ^«/'> . .^ „ ,„ 3-n. 5-n l + 2t-'2f-n ^,. „^ 



^' = . (n-1 . ^^osze.^r-^. ^^^ ^^_^^, .<p{t;t'), 



