The Interchange of Two Differential Resolvents. 89 



12. The above is an extension of Boole's Theorem. 

 For, make r=n—i, 0=1, £= — 1, and c= — 1 ; then (B) 

 becomes 



y n -xy n ~ 1 -\=0 (B ) 



which is the algebraic form dealt with by Boole, and (B') 

 becomes 



Aw _ "I o>i \n-l 



w[D] n «=N -— D + --1J (D-m-n)x n u . . (B' ) 



which is connected y with (B ), as Boole shews, 13 through the 



relation 



u = y"\ 



1 3 See Boole's paper " On the Differential Equations," &c. , cited in footnote 2. 

 See also a paper of mine "On Differential Resolvents," printed in the Pro- 

 ceedings of the London Mathematical Society, Vol. I., 1865, Paper IV. I may 

 add here that the explicit form of the complete cubic differential resolvent, 

 published in the British Association Report for 18S6 (pp. 439 — 443), was 

 calculated independently by Mr. Robert Rawson and myself. 







