182 Sir William R. Hamilton on the Argument of Abel, 



we shall then have, as the announced expression for this isolated term, the 

 following : 



the sign of summation here extending to all those terms in which every index 

 such as 7 IS equal to zero or to some positive integer less than a . 



i 1 



Thus, in the case of the function of second order b", which represents, as we 

 have seen, a root of the general cubic equation, if we wish to obtain an isolated 

 expression for any term f of its development already found, namely the de- 

 velopment 



b" = S .(b' . a/") = b: + 6/ o," + b.l ar = tj' + 1," + 1,", 

 I3,"< 3 \ /3." y 



we have only to introduce the function 



b"= E Jb' .p, .<' ) 



and to employ the formula 



t" =b' .o;^'"=i.s .fb" .pr^'""") 



ft" ft" r,"< 3 V r." / 



■ft", „ . -2^.", 



In particular, 



in which 





K'=b:^-b;p,al'^b^P,'ar, 

 b.."=b,'+b^p,^a^'+b^P*ar, 



and in which it is to be remembered that 



p3^-\- P3-\- 1 =■ 0, and therefore ^3^= 1 , 



