478 The Rev. B. Bronwin on the Inte^ation of 



which is equivalent to ^ 



sv^JlxJ) + A:)M=/(a:D) A, 



into which it is changed by changing e* into a. But this last is 

 only a particular case of the more jrencral theorem, 



a*J['jr+k)u:=Jl',r)x'u, (A) 



where 



w=afD + X(a7). 



Here \{x) denotes any arbitrary function of a?, and may be a 

 constant or nothing. This theorem is a modification of one 

 which I have given in a memoir printed in the second part of 

 the Philosophical Transactions for 1851. 

 A more general form is 



^{x)''f{'7r + k)u=flwmxYu, . . . (B) 

 where 



The first of (a) is easily verified in both these cases, as also in 

 the two that follow ; therefore the theorems (A), (B), &c. are true^ 



'D^f{'GT-k)u=f{7r)J)'u (C) 



if 



v='Dx+\{J)), or ^^zxD + XCD); 



where, of course, the arbitrary function X(D) may be a constant 

 or nothing. 



A more general form than this is 



^{ji)>'f{^-k)u=j{^mu)''u, . . . (D) 



where 



-(J-^)>+MI>).or.=.(^)+MD). 



These might be derived from (A) and (B) by the commutation 

 of symbols; that is, by changing x into D, and D into —x, or 

 by changing x into —I) and D into x, and by suitably changing 

 the functions /, X and </>, when necessary. 



In order to apply these theorems to the integration of linear 

 difi'erential equations with integer functions of x for their coeffi- 

 cients, suppose 



\{x) = a + fl,.r + flfgX^ + . . . . , 



and X(I)) a function of the same form ; we can put any equation 

 under the form 



X=/{7r)u+f,{7r)xu^M7r)x^u+ (1) 



For 



ITU = xDu -h\(x)u, 



