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



r f^^l r (-\ — ' 



\2J \2) 1.3.5 2s + lb''+'+" (l-af)' fx'^^'dx 





1+w ^ ^•'^+'+" ^o' 



2 



J {\-x) 



Avhere after the integrations have been effected, x ought to be made 

 equal to - . 



The value of the integral last found being substituted in the expres- 

 sion immediately preceding, and the finite integral taken relative to s 

 from * = to * = X gives for the repulsion of the inner sphere. 



a ^ ll+n\ f2 — n 



¥~J V'~2 



) 



-2.7i.n + 2 n + 2s-4> (by"(l-x')'' raf'^'dx 



''^" 2.4.6 2s [r) ic^'+'-^" J„ 



(1-x^y 



.2\2 



-47rv/,rP^a^/-" » n-2.n.n + 2 w + 2^-4 (a^ rj..^„ /, _^.^\ 



„fl+n\ l2-n<:^' 2.4.6 2s (rj ■'"'^^'^ ' ^* ^' ' 



i ) [ 2 J 



since F (^) =\/ir, sin ( tt] = 



^'D^C-?)' 



and as was before observed, a; = - . 



a 



But we have evidently by means of the binomial theorem, 



/ _ ff-j;-\i^ _ . n-2.n. w4-2 w + 2^-4 /«^y, 



I rW ' "" ° 2.4.6 2* \r] ' 



and therefore the preceding quantity becomes 



(15) 7^^ L dxaf 1 ' (l-x^)" . 



