Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 59 

 and consequently, 



sin (^.) 



. (n-\ 



2 sin I —7: /r I 3_^ ,2 ,4 





;ii^^«--(«=-ir.''(i-5)"' 



•* sin 1 ^ "I B-B „M 



q{a'-V)—{a'-r'y\-, 



the particular value -4'°^ being one half only of what would result from 

 making i = in this general formulje. 



But 4'' = evidently gives £^''^=0, and therefore the expansion of 

 f{a!, y') before given becomes 



fix', y') = J^'^ + A^'^ cos ff + A-'^ cos 20' + ^''^ cos 30' + &c. 



= 1-^ -g(«^-l) ^ («=-r'0-'.{| + - cos0'+ -cos20' + &c.| 



or by summing the series included between the braces, 



. (n — 1 \ 3-» 



JKx,y)- -^ ^a^-2«r'cos0' + r"' 





sin (^ .) 



Q 



iJ being the distance between P, the point in which the quantity of 

 fluid q is concentrated, and that to which the density p is supposed to 

 belong. 



Having thus the value of /(a;', y') we thence deduce 



(n — 1 'N 3-, 



p = (1 - /»)-/(x', y') = - — i-| i (1 - /') ^ ? — ^. 



sin 1 — ::: — TT I „_3 (a^ — Yy^ 



H 2 



