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



The preceding method is directly applicable when the forces X, Y, Z 

 are given explicitly in functions of x, y, x. But instead of these forces, 

 we may conceive the density of the fluid in the exterior bodies as given, 

 and thence determine the state which its action will induce in the con- 

 ducting sphere A. For example, we may in the first place suppose 

 the radius of A to be taken as the unit of space, and an exterior con- 

 centric spherical surface, of which the radius is a, to be covered with 

 fluid of the density U"^'^: ZJ"'"* being a function of the two polar co- 

 ordinates 6" and ■zsr" of any element of the spherical surface of the same 

 kind as those considered by Laplace {Mec. Cel. Liv. iii.). Then it is 

 easy to perceive by what has been proved in the article last cited, that 

 the value of the induced density wiU be of the form 



p = [/-'Wr'' (1 - r"'y' .f{r") ; 



r', &, -ar' being the polar co-ordinates of the element dv, and £/'<'* what 

 Z7"<'> becomes by changing Q", -sr" into 9', tst'. 



Still continuing to follow the methods before explained, (Art. 4. and 5.) 

 we get in the present case 



f{af, y', «') = t7'<Vy(r'^) =/«, 

 and by expanding y(r'^), we have 



/(r'^) = i?o + B,r" + B,i'' + B,r" + &c. 

 Hence, /'" = B,U'^\ and 



' . ln-% \-"'^'''^ 2.4.6 2^-2^ ^ 2.4 2^' 



sm(-^.) 



n-l.n + 1 w + 2? + 2if-3 



^3.5 2«-l-2«r + l ■ 



Then, by giving to t all the values 1, 2, 3, &c. of which it is sus- 

 ceptible, and taking the sum of all the resulting quantities, we shall 

 have, since in the present case V reduces itself to the single term V^\ 



sm (-^.) 



n-l.n + 1 n + 2i + 2t'-3 ^ 



^ S . 5 2i + 2f + l ' 



the sign S belonging to the unaccented letter t. 



