and T7-ansformation of certain Differential Equations. 9 

 is given as being y = (D^ -\-k'^)P n, 



where (D2 + ^')« = 0, 



a result wliich is evidently identical with j/ = 0, which is in- 

 deed a solution of (I.), but not the solution of which the author 

 is in quest. 



The error consists in inferring the equation 



from 



The deduced equation ought to have been 



D^xti+{2p-i)Du + k^xu={D'^ + k-)-^0 = ai>\nkx + bcoskxt 

 which will lead to the result 



more conveniently written 



3/ = (D2 + Fy'-'|i(DHF)-/'.o}-. . . . (2.) 



In solving differential equations by successive operations of 

 this nature, a difficulty frequently occurs with reference to the 

 introduction of constants. Thus every operation denoted by 

 (D^ + F)-' introduces two constants ; and in many cases all 

 the constants thus introduced except two, which are arbitrary, 

 must be determined in terms of these two arbitrary constants 

 by reference to the original diflferential equation. 



This difficulty does not occur in the above equation (2.), 

 which may be written in the simpler form, 



j/= (D2 + F)/'-i <[ i (asin/t.r + bcoskx)\. 



If p = l, 1/= - {asin kx + bcoika). 



2 2k 



If p = 2, ij =— ^ (asiii/t-jr + ^'cos^-o:') + — ^ (6sin/f.r — «cosA\r), 



&c. &c. 



It will be found, in like manner, that the solution of the 

 second of the equations given by Mr. Bronwin, viz. 



rf^V , dy 

 is y= (D + /f .r)'' {D-'(D-f /a-)-(/'+')0}, 



