Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 49 



But whatever the density P on the inducing spherical surface may 

 be, we can always expand it in a series of the form 



P = C7"<°>+ Z7"<'>+ C7"<^>+ Z7"® + &c. in inf. 



and the corresponding value of p by what precedes will be 



„ . /n-Q, 

 2sm 



P = - 



<a{a'-l) ' .{a'-r")-H^-r") 



X { t7'W+ [/'<')- + t7'<='^ + f7'<^>^ + &c. in inf.] ; 



Ijm^ U'M^ jjm^ &c. being what U"^'\ U"^'\ U"^% &c. become by changing 

 d", w" into ff, Ts-', the polar co-ordinates of the element dv. But, since 

 we have generally 



^d&'d-uy" sin 6)"PQ« = fdff'd^" sin 6" C7"<"Q» = ^^ C7<", 



{Mec. Cel. Liv. iii.) the preceding expression becomes 



-sm(-^.) 



p = _> a{a'-\) ' K-r'^)-'(l-0 ' jd&'d-sr" sin &'. 



2:(2e + l)PQ«^; 



a* 

 the integrals being taken from 0" = O to 0" = 7r, and from ■bt" to ■sr" = 27r. 



In order to find the value of the finite integral entering into the 

 preceding formula, let R represent the distance between the two ele- 

 ments dff, dv ; then by expanding -^ in an ascending series of the powers 



r' 

 of — we shall obtain 

 a 



— = ^ _ 2°°Q<*>.— -, 



B Va^ - 2ar' [cos 0' cos 0" + sine' sine" cos (-ar' -•23-")+/* ° *«" 



Mec. Cel. Liv. iii.). Hence we immediately deduce 



^ = .r«»e^, and .^4,^^ = K(^.>1)«?»^. 

 Vol. V. Part I. G 



