[From the Quarterly Mathematical Journal, vol. n. (1858), pp. 167—171.] 
Lagrange has given the following formula for the sum of the inverse n th powers 
of the roots of the equation x = u + tfx, 
2 (z~ n ) = u~ n + (— nu~ n ~ l fu) J + (— nu~ n ~ l fhi)' + &c. 
where « is a positive integer and the series on the second side of the equation is 
to be continued as long as the exponent of u remains negative (Théorie des Equations 
Numériques, p. 225). Applying this to the equation x = 1 + tx?, we have 
2 (ar n ) = 1-” - j 1.1-’*+*-* + —A~_ 2 | + — t-. .. 
+(-)* »(—a» + g-^-9» + i) „. j-»,,»-, _ &c . (2) 
1.2...q w 
to be continued while the exponent of 1 remains negative. 
Let n = ps + p, p being not greater than s — 1, the series may always be continued 
up to q = p, and no further. In fact writing the above value for n and putting 
q = p + 6, the general term is 
(~Y +6 f,2.~.T(ya~+ 9) ^ + ^ ( P ~ 0S + ^ + 6 ~ 1 ) ' * * ( p “ 6s + 1) l _,p_ôs+ '‘ +0, . 
Now if p + p — 6 (s — 1) is negative or zero, the term is to be rejected on account 
of the index of 1 not being negative, and if this quantity be positive, then since