Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 43 



T X* 



If HOW we make x — — , the same quantity may be written 

 (16) , . , -; tx'dxiX-x^)' 1 —\'- 



Having thus the vahie of the repulsion due to the inner sphere B 

 on an exterior point p, it remains to determine that due to the fluid 

 on ^'s surface. But this last is represented by 



, 2 7raP^ (L_ {a-\-rf-''-{a-rf- 



' l — 7i.S — ndr r 



{Mec. Cel. Liv. ii. No. 12.) Now by expanding this function there re- 

 sults 



1~ ■ r "*" 4.5 ^ "*" 4.5.6.7 



^TrP'a'-'r. ^^ ■ U + "T l^ 2'- + """,T^":" ' " .g^+^cl 



. r./ i» 2-w .^, ra.w + l.?« + 2 /i + 2*-l, ^.r" 



= 4.PV-V.-g-.2„ ,.5.6 ,,^3 (^-^1)^- 



The last of these expressions may readily be exhibited vinder a finite 

 form, by remarking that 



flx"dx{l-a^) ^ (l - -^) ^ =/lx''dx(l-x') " S ^^ g ^^^ .-^ 



/ 2^ + w + l \ / 4-/A 



«.?f + 2.;? + 4 w + 2jf-2 y^' V 2 / V 2 / 



~ " 2. 4 .6 2s 'a^'' ^^l2s + 5\ 



2 r 



r {lzl\ r fl+^^ 



V 2 / \ 2 I 2-n ^^ n.n + l.n + 2 n + 2s-l , ,.r^' 



© 



3 ' " 4. 5 . 6 2* + 3 ' 'a 



Hence, since r (^) = v^tt, the value of the repulsion arising from ^'s 

 surface becomes 



F 2 



