Ma GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. Sfi 



We will now consider more particularly the part of V due to any 

 of the terms in f as -<4^'^ cos i& for example. The value of this part 

 will evidently be 



r r /dr'dff{\ - r"fA^^ cos iff 



{r'-^rr' cos (Q - ff) + r'')~^ ' 



the limits of the integrals being the same as before. But if we make 

 6' = 9 + (p, there will result dff = d<p, and cos i9' = cos id cos e0 — sin iO sin «0, 

 and hence the double integral here given by observing that the term 

 multiplied sin i<p vanishes when the integration relative to (p is effected, 

 becomes 



cos ie/lA^'^r'dr' (1 - ry f^ ^"^ ^"^ ''I' —^ ; 



° {r'' — 2rr' cos (p + r"^)~^ 



If now we write F"/*^ for that portion of V which is due to the term 

 «/*^r"+^* in the coefficient A^'^ we shall have 



r,» = «/'> . cos ieflr^^'^^'dt" (1 -ry /" "^"^ ^"^ "^ ^ . 



" {r^ — 2rr' cos (p + r"^)~^ 



But by well known methods we readily get 



•^'^ d(p cos i(}) 



L 



{r' — ^rr' cos (f> + r'^) ^ 



i ^i-.-iv» ^"' n-l.n + 1 n+2t'-S n-l.n + 1 n+2i+2t'-3 



-2irr.r i„^„,. 2 _ ^ 2^ "" 2 . 4 2i + 2t ' 



when r'>r, and when /<r, the same expression will still be correct, 

 provided we change r into r' and reciprocally. 



This value being substituted in that of Fj*'' we shaU readily have by 

 following the processes before explained, (Art. 1. and 2.) 



F,w = 27ra/'V* cos 10 2o r ' — r 7 -—p 



P^^ . ,.„ ,S + 2t-2f~n\ 



»-l.» + l « + 2« + 2#'-3 "^f^ ' I \ 2 



X — ^:: -. 7;r- — :;^-r, X 



(/3 + i)r[^ 



2 . 4 2i + 2/' ( 2li + 5 + 2t-2 f) 



^^[ 2 J 



