190 The Rev. B. 13ronwin on the Integration of 



If rm=p — 1, we shall find by substitution and reduction, 



tvk , v (vr+p-m) .... (ot + ct + 1) -^ 

 D u l +ku 1 =X l = - ..,, {zT -p+l) ' 



and 



« 1 - SS (D-+*r i x 1 . 



These are the two cases of the proposed equation considered 

 by its author. To take a case not noticed by him, change 

 p\p — l) into « 2 ; then the equation becomes 



x*D m u- ft 2 D m_2 M + kx% = 0. 

 In this case make X(D) = 1 , and the transformed becomes 



w 2 (D' n + k)u -n 2 D m « = D 2 0, 

 or 



(OT-«)(OT+n)D' n M + ^ 2 M=D 2 0. 

 Here assume 



_ (■or — m)(ot — n + m) . . . (ct— n + (?'— l)m) 



•cr(OT + m) .... (V+(r— l)w) 

 aud we shall have by (C) 



(g-» + »z)....(g-n + r»t) 

 D M== (<* + >«) .... ( W + ri«) D "'• 

 Make rm — n, and we find by substitution and reduction, 



and 



Ml =(D m + A:)- 1 X 1 . 



Perhaps by assigning other forms to \(D) we migbt obtain the 

 solutions of other cases of this equation. 



An example of the use of the arbitrary function \{x) shall now 

 be given. 



Let x*I)% + 2aBu+(b+^p\u = 0. 



Here, according to the former paper, 

 7T=a?r)+X(a'), 7ru — ajDu + \(x)u, 

 ~Du=x~ l 7ru—%~ 1 \(x)u, 

 I>%=x- z 7r q u-x- 2 (^l+2\(x)')7ru-x- 2 (\(x) 2 +\{x)+x\'{x)')u. 



If we substitute these values in the proposed equation, we 

 shall find that by making \(x) = — the result will be reduced to 

 the very simple fomi 



7T 2 M — 7TU + bu = 0, 



