188 On Differential Equations 



This result coincides ultimately with Boole's. For if in (xv) 

 we replace y by y~^ we have an equation employed by Boole 

 {ibid. p. 736) and if in (xiv) we replace 2/ by u we have its 

 n-sjry resolvent as given by him {ibid. p. 737). 



25. If we make 



Q = ^B -^— — 1 - - - (xvi) 



n n ^ ^ 



and 



„ ^^ ('^ ^) ^ / --x 



H = ^ + n — r - (xvn) 



n ^ ' 



the general theorem may conveniently be expressed as fol- 

 lows : — The 7i-ary, or Boolian differential resolvent of (v) is 



[D]^ u — [(?]'■ \Hf-^ x''u = - (xviii) 



26. I communicated the generalization of Boole's propo- 

 sition to Mr. Harley by the last October mail, together with 

 a verification — both verifications, I believe. By the last 

 December mail I received a letter from Mr. Harley, dated 

 Oct. 17, 1865, which therefore crossed my letter to him, and 

 in which he makes a very near approach indeed to the true 

 generalization : so near, indeed, that, inasmuch as the over- 

 sight into which he has fallen could not long have escaped 

 his notice, the generalization may be regarded as having 

 been independently made by me in Queensland, Australia, 

 and by him in England. By the last (February) mail I 

 received from Mr. Harley a letter dated Dec. 18, 1865, in 

 which he acknowledges my letter of the 18th Oct. and in 

 substance says that the true generalization is 



[PY u — (—y-^ [Gf [J7]^-^ x''-^u = - (xix) 



where 



H = If — 1 - - (xx) 



n n ^ ^ 



Now, inasmuch as 



^ ^—+n—r\ =\ D 1 (-1)«-^ (xxi) 



Y n J in n \ ^ '' ^ ■' 



the only real discrepancy between our results consists in the 

 different indices of x. Mr. Harley 's factor ' x'"'^ ' seems to 

 me erroneous, and neither to follow from the law of deri- 

 vation of a differential equation from a series, nor to agree 



