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



If now V represents the value of V due to the exterior surface, 

 it is clear from what Laplace has shown, {Mec. Cel. Liv. ii. No. 12.) that 



^ = !y^ = (3^:^ K«+^r"-(«-'-M; 



rfo- being an element of this surface, and g being the distance of this 

 element from the point p to which V is supposed to belong. 



If afterwards we conceive that the function V is due to the fluid 

 within the sphere itself, it is easy to prove as in the last article, that 

 in consequence of the equilibrium we must have 



V +V= const. 



But V and consequently V is of the form F^"', therefore by employ- 

 ing the method before explained, (Art. 4.) we get 



/(ar', y', %) =/'(»' =/„("> +/(">. r'' +//>. r'' + &c. = B, + B,r'' + B, r" + &c. ; 



where, as in the present case, ^''°>, yi'<°', ^''% &c. are all constant 

 quantities, they have for the sake of simplicity been replaced by 



J?o, jBi, B.^, &c. 



Hitherto the exponent /3 has remained quite arbitrary, but by making 



/3= — -— ^ the formula (11) Art. 4. will become when « = 0, 



ir(o)_o 7? < 2y V 2 ; ^ ,,, 4-».6-w 2t-9.t' + ^-n 



^' -2'^^'- YW) ' 4 . 6 2^-2^+2 



n-2.n-l « + 2^- 3 



"^ 2.3.4 2^ + 1 



{/i-2)Tr'Bt ^ ,„ 4-W.6-M 2^-2/' + 2-« w-2.«-l « + 2/-3 



2.?^ . ^ -r^ ^w . n ^ 



. (71-2 \ • 4.6 2^-2^ + 2 2 . 3 2^' + l ' 



sm(— .) 



Giving now to / the successive values 0, 1, 2, 3, &c. and taking 

 the sum of the functions thence resulting, there arises 



r= F<°' = Fo<"> + rr + T^-P + ^3<°* + &c. = s. r/"> 



(«-2)7r'^ ^P^ ,,, 4-W.6-W 2t-2f + 2-n n-2.n-l «+2^'-3 



"""T^ri"^ ' 4.6.8 2^-2^' + 2 "" 2 . 3 2/' + l ' 



sm (^.) 



where the sign S is referred to the variable t and 2 to ^. 



