Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 55 

 and afterwards by making 



ro = ro» + r/'> + r.» + r « + r« + &c. 



we shall have 



TT* 



V^'^= , r* cos i6 into x 



/« — 1 

 sin 



sin (-^vr) 



«<,'>.1.0(«;O) 



+ a<'>.?^.<^(e;0) + «?>.!. 0(«; 1) . r= 



+^'- 2.4.6 •<^(^;o) + «^"-2r-^— 0(^;i)-^ 



+ «^^^.0(«; 2).r' + af.l.cj>{i;3).f^ 



+ &C +&C +&C +&C 



Conceiving in the next place that F is a given rational and entire 

 function of x, y, the rectangular co-ordinates of p, we shall have since 

 X = r cos 0, y = r sin 0. 



{25) r= C<") + C('> cos 6 + C-'^ cos 20 + C('> cos 3 + &c. 



+ ^« sin + ^('' sin 29 + E^'^ sin 30 + &c. 



of which expansion any coefficient as C^'> for example, may be still 

 farther developed in the form 



C« = ""'-^ {di\(p{i', 0) + c>{K(p{i; l).r'+4^.(p{i; 2).r' + ke.}. 



sm (-^ .J 



Now it is clear that the term C> cos iO in the developement (25) 

 corresponds to that part of F which we have designated by F''', and 

 hence by equating these two forms of the same quantity, we get 



F» = Cw cos ie, 



