14 Mr green, ON THE LAWS OF THE EQUILIBinUM OF FLUIDS. 



acnd the part of U, due to the general term i?(/-''^' in (/*), will be 



(J) r'fd&diir' sm 9' Q^Bt/lr'^-"^^'-' dr' {l-ry-, 



which, by employing the formula (3') Art. 1., and rejecting the inad- 

 missible terms, gives for truncated formula 



[ 2 j 



By continuing to follow exactly the same process as was before 

 employed in finding the value of Fl, we shall see that * must always 

 be an even number, say 2 s'; and thu« the expression immediately {Br- 

 eeding will become 



,, l4!-n + 2t-2s 





^ (6-n + 23 + 2t-2s' 



2r I 



; — r'^' fdO'dw sin d'^2.- B,. 



2 J 



Moreover, the value erf V^ will be 



r, = u: + u: + u: + u: + &c. ; 



U^, Ui, Ui, U3, &c. being what Uo, Ui, Ui, &c. become when we use 

 the formula {b') instead of the complete one (h). 



The value of V answering to the density 



p = p, + p, = (l-ry. /{:>/, y',z'), 



by adding together the two parts into which it was originally divided, 

 therefore, becomes 



r = r,+r,= t: + t: + t: + t/ + &c. 



+ £/■„' +t4'+C7;'+t7e' + &c. 



When /3 is taken arbitrarily, the two series -entering into V extend 

 in infinitum, but by supposing as before. Art. 1., 



— n n 



