48 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



By equating separately the coefficients of the various powers of the 

 indeterminate quantity r, we get the following system of equations : 



^ . (n-2 \ 



2 sm — — — TT . AC 



2 . 4 



^ • fn-2 \ 



, I o -n Tk 4 — w „ 4i — n.6 — n . 



TT 2 2.4 



^ . fn-2 \ 

 2sm(-^.) 



TT 2 2.4 



&c &c &c. 



But it is evident from the form of these equations, that if we make 

 generally ^,+i = a'^Bt, they will all be satisfied provided the first is, and 

 as by this means the first equation becomes 



2 sm — -— TT , A AC 



Tr»/, 4 — « „ 4 — M.b — « . . \ 

 «-»- = J?„ (l + -g-«- + 3 ^ a-^ + &c.) 



= J5o(l-«-'^) ^ = .Bo«*-"(a'-l) ^ . 

 there arises 



„ . /M-2 



2sm 



(n-2 \ 



^^ La-'-'{a'-l)' , B, = B,.a-\ 



Bo = -— -a-'-'ia'-l)' , B, = B,.a-\ B, = B,.a-\ !>ic. 



TT 



Hence 



f{r") = B, + By' + B,r'' + &c. = ^„ (^1 + ^ + ^ + &c.) 



»8 2 sm I — — - TTJ ^_„ 



= ^„fl_ !l)-i = ^„a=(a^_/^)-'= L* ia'+'(a^-l)~(a^-r'^)-\ 



\ a I If 



and the required value of p becomes 



(21 ) (D = C7'< V; (1 - r'^)^/(r'0 



2sm(-^.j 



(a' - l)'^"a i7'» f^') '(«= - /=)-' (1 - r"y\ 



