and on Go-resolvents. 187 



in which D is an operative symbol defined by 



and the notation 



\cCf = a (a — 1) (a — 2) . . (a — 5 + 1) - (viii) 



is adopted. This theorem is an extension of "Boole's pro- 

 position at pages 734 and 735 of the Philosophical Trans- 

 actions for 1864. If we make 



T = n — 1, x = e^ 



d_ 

 cl e ' 

 then (vi) will become 



and therefore regard D as representing the operation 



\DYu+ D+——1 I 1 \e''^u = (ix) 



•- -• L n n j ^n n J ^ ^ 



and so coincide with the equation given by Boole at the 

 pages cited. 



24. As a second verification of this theorem let r = 1, 

 then (vi) becomes 



1 .n _. ....>^_^^— ^n,.o__il 



71—1 



\mn ^^ rj) ^ ^ ^,^M J) ^ ^^ J aJ'^'i6=0 (x) 



•- J 71 ^ ^In n J ^ ^ 



which, when m = 1, becomes 



W'y-i (^-^'^+ l)[r,- ,, D + n-l J «»2/=0 (xi) 

 or, which is the same thing, 



which is the 7i-ary differential resolvent of 



y'' — xy — 1=0- - - (xiii) 



Now if in (xii) we replace o) by — x, and afterwards, as be- 

 fore, substitute e^ for x, then (xii) becomes 



which is the 7i-ary differential resolvent of 



y'^ + xy — 1=0 - - (xv) 



