192 On Differential Equations 



Addendum to the foregoing Paper. 



Under date Oakwal, near Brisbane, April 9, 1866, Chief 

 Justice Cockle communicated to Sir Redmond Barry the 

 following additions to his Paper " On Differential Equations 

 and on Co-resolvents." 



I. We may readily deduce a differential co-resolvent of 



xy''—y^+z = 0, - - - - (a) 



from which we obtain successively 



y»=z + xy"" - - - - -(b) 



y =z (z + a?2/«)« - - - . (c) 



y'''= {z + xy'')^ - - - -(d) 



Hence, by Laplace's theorem, 



II. Now since the general term of the series for 7/"* is 



m d'-'^ f ^ '»-i) X*- 

 a dz'-^X jl.2..r 



or 



m d"-^ ( ^^'1^-1 1 X'' 

 a dz'-'-\ ) {rY 



or, if we make z = l after the differentiations, 



m \r 11 m ^ ]''"^ x^ 

 ~^Vi: "^ a —J \rY 



we may put 



nfi\rn TYi 

 a I a a 



r -,^ III, r lb . 'iru ^ 



>-i 



(f) 



and, consequently, 



\r — a^u — — ^ '- 1 



L ^ r-a a i a a J 



