4G6 report - 1878. 



13 On Certain Linear Differential Equations. By the Rev. Robert Harley, 



F.B.8. 



By an ingenious application of Murphy's theorem, Mr. Eawson has shown that 



if u.„ = — 2y", the summation extending to any numher of the roots of the alge- 



n 

 hraical equation 



y m + ay r + bx = o. . . . (a), 



y feeing considered as a function of x, then 



dx a(m — r) ax m 



du„+ m _ br ,du. 



, du H n s , -s 



dx m — r dx r 



Mr. Rawson has also shown how the " differential resolvent " of (a) may he calcu- 

 lated hy means of the equations (/3) and (y), when particular values are assigned 

 to m and r. The object of this paper is to show, first, that Mr. Rawson's differen- 

 tial equations may be readily derived from the algebraical equation, without the aid 

 of Murphy's theorem, by a process suggested by Professor Oayley ; and, secondly, 

 that Mr. Rawson's results may be generalised and presented in a compact symbo- 

 lical form. 



It will be convenient to deal with the equation 



ay m + by'' + c x = o. (1) 



which is unaltered when a, m are interchanged with b, r. Differentiate (1) with 

 respect to x, and multiply the result into y", then 



(amy m+n ~ x + bi-y r+n ~ r ) y' + cy" = o, 

 where differentiation is denoted by an accent ; but 



my n ~ l (ay m + brf + ex) y' = o; 

 whence, eliminating y m + n ~\ we have 



b (m - r) y r+ n_ y + cmxy n ~h/ — cy n = o, 

 which, summing for any number if the roots of y, gives 



b \m — r m-r "/ 



Interchange a, m with b, r ; then 



M ' ra+n =! <L (_!!_»'„- _=_„„). (3) 



a \r— m r—m J 



These results, viz. (2) and (3), agree with Mr. Rawson's equations (/3) and (y) 

 They may be otherwise written 



Dm ,. +)1 =- ^(jLD-J'-k. (4) 



b \ m—r m — r) 



m n+m = - t x (-Z- D - -!-)„„ (5) 



BT a \r-m r — m/ 



where D = .*'-r-* 

 dx 



Adopting the usual factorial notation, viz. : — 



[<9] a = <9(<9-l) (6-2) . . . [6-a + l), 

 according to which 



a + S a 



[ff] =[<9] [<'-«], 

 and effecting reductions by the aid of the theorem 



/ (T))x a u = x'f (D + a)u, 



