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



which by substituting for F"*'' and C^ their values before exhibited, and 

 comparing like powers of the indeterminate quantity r gives 



/> , 1-^ 3 — n,. 3 — n.5 — n,i. 3 — n.5 — n.7 — n,., , 



2 

 &iC.— &c &c 



of which system the general type is 



C<'> = (1 - e)~ . «« ; 



the symbols of operation being here separated from those of quantity, 

 and e being used in its ordinary acceptation with reference to the lower 

 index u, so that we shall have generally 



f.m „(i) _ ^ (0 



The general equation between «!'' and cll^ being resolved, evidently gives 

 by expanding the binomial and writing in the place of eci'', e''&i\ ^c'i\ &iC. 

 their values c„*j\ , cj-i\, Cu%, &;c. 



(26) ««=(l-e)^c« = c<;>+^c„« + ''~^-''~^ 



2 — ' ■ 2 





... « — 3 . ra — 1 . w + 1 (i) , s 

 %+ 2.4.6 "-- + ^"- 



Having thus the value of af we thence immediately deduce the value 

 of ^<'' and this quantity being known, the first line of the expansion 

 (25) evidently becomes known. 



In like manner when we suppose that the quantity J5^'> is expanded 

 in a series of the form 



j5:« = — ^TTT ^^»"- *^ (^' ' 0) + ^'* "^ (^' ; 1) • ^' + ^^'* <^ («■ ; 2) . f^ + &c. ^ 



sin 



sin(^.) 



