Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 47 



If now V represents the function analagous to V and due to the 

 fluid on the spherical surface, we shall obtain by what has been proved 

 (Art. 3.) 



V = WK 27r«^ W'^ ^'7^ /-i(/M {i) (f-2artx-V a'i^', 



X • ^ ,Ot« • • • t 



{i) representing the same function as in the article just cited. 



Moreover, it is evident from the equation (10) Art. 4, that 



,. ,j ,., ,^ ^ , ,.^ ^,1.2.3 i ^n-l.n + 1 n + 2i + 2t'-S 



/ild^{t){r^-2ar^ + a^)^ = 2«'-" ^ 3 ^■_^ 2 ^ ^ ^i + ^ TTT 



n-2.n n + 2t'-4> IrV^^^' 



"" 2.4 2^ \a) ' 



and consequently, 



(.9) r'=cr<o.w-.. '';';''^';;;;;'-^f;^;:^7^ 



«-2.w ?« + 2if'-4 /r\' + '' 



2.4 2^' \a) 



the finite integrals extending from t' = Q to t'=<x). 



Substituting now for F' and ?^' their values in the equation of equi- 

 librium, 



(20) const. = r'+ f; 



we immediately obtain 



const. = i7".47rflr' ".2 



3 . 5 2i + 2t' + l 



n-2.?i w + 2^'- 4 //•\'+''' 



2 .4 2f 



Q 



^ 2^" rm 97? v,.<+2,' ^-1-^^ + 1 « + 2i + 2^'-3 



^ . fn-2 \ ' "3.5 2» + 2r-l 



sm(-^.) 



n-2.fi Ti-{-2t'-4> 4-W.6-W 2^-2^' + 2-« 



"^ 2 .4 2? ^ 2 . 4 2^-2^ 



the constant on the left side of this equation being equal to zero, except 

 when i = 0. 



