THE EQUILIBRIUM OF FLUIDS. 175 



we shall have 



; cos id into x . . 



sn 







2 . 4 

 3 n.5 n.l n 



&c + &c + &c., 



Conceiving in the next place that V is a given rational and 

 entire function of x, y, the rectangular co-ordinates of jp 9 we 

 shall have since x r cos 0, y = r sin 6, 



V= (7 (0) + <7 (1) cos + (7 (2) cos 20 + <? {3) cos 3(9 + &c. 



+ E (l) sin + E sin 20 + E (S) sin 30 + &c ....... (25), 



of which expansion any coefficient as (7 (0 for example, may be 

 still farther developed in the form 



Z -; 2) 



Now it is clear that the term (7 (i) cos iO in the developement 

 (25) corresponds to that part of V which we have designated 

 by F (i) , and hence by equating these two forms of the same 

 quantity, we get 



which by substituting for V (i) and C (i) their values before ex- 



