108 The Rev. B. Bronwin on the Integration of some 

 Therefore 



(A + AiO? +A^_ia?''-'). 



We must now put aD' v^ — 1 for k, and change A, Aj, &c. 

 into functions of j/. Then 



,2.u^-. A = e-^-'-^^'f{7/)=f{>/-2ax), &c., 

 and 

 ^-(0- + k-) u = ^"■'D' T>-!' [f{y-2ax) + .r/, {ij-2ax) 



-\-xP-'fp., (j/-2ad?)} =£«'-°' {cp{y-2ax) + a:<^^{y-2ax) .... 



+ xP-^ ^p_, (^-2a.r) +^I/(^) +a.-vl/i( j/) +xP-' 4//,_, (i/)}, 



where \J/ (3/), \|/, (3/), &c. are thearbitraries of the integrations. 

 And here I must observe, that the functions denoted by <p, 

 <Pp 4'j 4'u &c. are supposed to be changed when necessarj', 

 although the same symbols are still retained to represent 

 them. 



Operating upon the last result with 



(D-^ + F)P-^r-i =(D + aD')^-'(D-«D')''-'^-' =s-"'^ 



we have 



-VxP-^ ?2p-2(j/-2a.r) +a;-''vj'(3/)} = e— i>'D''->{.r-/'^(^) 



-f a:-''+»^,(j/) +A'^-2<p2^_2 {y) + x-P^y + 2ax)) 



= x--P+' [<^{y-ax)-\-^y-{-ax)] +x--P+''{(^^{y -ax) 

 + -^y{y-\-ax)} +x-P{fp^^{y-ax)+-^p.,{y + nx)]. 



Substituting this value in the given equation, we find (p,, f^^ 

 &c., and \|/„ vt'^f &c. to be the successive differential coeffi- 

 cients off and \I/ ; and ultimately 



z=a?--^'+'{<p(j/ — a^') + 4'(i/ + o.r)} +ax--P"'-{f\y — ax)—^' 



{y + ax) } + o2 1^ ^-2p+3 { f "(y - a*') + ^"{y ^ax))-^ &c., 



to the term containing x~p inclusive. 

 As a second example, let 



Or if ^ be put for D', 



