Mn GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 39 



Again, by substituting for V and V their values in the equation 

 V^ + V= const, and expanding the function V we obtain 



,/^*' ti-2.n-l.n n + 2f-3 



const. =47rP'«'-".2 



«^"" 2 . 3 .4 2^ + 1 



(w-2)7r' „ „ ,, /4-».6-?? 2^-2^'+2-w n-2.n-l ti+2t'+3 



"^ (n-2 X '^^^''' 4 . 6 2f-2f + 2 ^ 2.3.4 2^ + 1 



sm 



..) 



2 



which by equating separately the coefficients of the various powers of 

 the indeterminate quantity r, and reducing, gives generally 



(n-2 



2P'«'-"-"'. sin /• 



TT 



2 I „2 — «.4 — « 2s — n 



-^ 2 . 4> 2s ^''^'-' 



Then by assigning to t' its successive values 1, 2, 3, &c. there results for 

 the determhiation of the quantities B^,Bi, JB^, &c. the following system 

 of equations, 



2P' „ . (n-2 \ p ^ 2-n „ 2-n.4<-n „ , » 



2P' . fn-2 \ „ „ 2 — « „ 2 — ;«.4 — ra ^ . 



=^ «'-".sin — r-TT .«-^ = P,-f — r- P. + — ^— r— P3 + &C. 



TT V 2 / 2 2.4 



2P' , . (n-2 \ ... „ 2-M„,2-ra.4-/< 



a'-", sm ( -^— t] • '^ = P.' + —^ -^^ + ~~;> 4 — P4 + &C. 



&c &c &c &c 



But it is evident from the form of these equations, that we. may satisfy 

 the whole system by making 



B, = B^.a\ B, = B.a-\ B, = B,,(r\ B, = B^.a-M>ic. 

 provided we determine Po by 



2P' . /n-2 \ „ ,, 2-n , 2 — n.4>-n ... . , 



= P„(l-a-)' 

 „ 2P . (n-2 \ . , ,,^ 



n-2 

 ,-2\ 2 



n — 2 

 Hence as in the present case, ^ = —^ , we immediately deduce the 



successive values 



2P . fn-2 



