Stfi Mw MURPHY'S THIRD MEMOIR ON THE 



Hence, ^=2cos0.— -X^*^ 



pa pa po 



= cosln+l)9 -{cosin-l^e + cos(n-S)e + ...+cos29+i}, 



and by a repetition of the same process, 



O Q . ,■> .*,{ ,-; . •-:.. 



— ^ = cos(»+2)0 -{cbsw0 + cbs(«- 2) . + ... + cos30 + cos^}. 



pa n-TXt 



Hence the laws by which S^-x and S^ are expressed are uniform, 

 and therefore we get for the required unknown quantities, 



2 2 2 _ 



po = Pa, pi=--pa> P8=~^Po P—i -P<>> Pn-Po- 



The positive values may be taken for the repulsive and the negative 

 for the attractive parts of the fluid, and if E denote the excess of the 

 former, we have 



[n n n n n] n [ n } n 



.-. po=»^-T — > which gives the complete solution of the problem. 



Thus the application of a process purely algebraical, conducts in this 

 instance to a transient function, for if we suppose the final and equal 

 densities po, p^ to be finite, all the intermediate values of the densities 

 p^, pa pn-x become indefinitely small when n is made infinite; yet 



they are not to be rejected, for if so, the total charge would be 4nra^, 



it 



whereas its actual value is only inra ~, an infinitesimal of the second 



pa 

 as Its itutuiu vaiuc is fJiii-y •±'iru, — 



order. 



.Vo 



Pm/ 



