188 EXAMPLES OF THE CHANGE 



Hence we infer that x -?- + y -~ is equal to nv n together 



with two series ; and by uniting like terras in the two series 

 we obtain a single series of which the general term is 



l-r+l) _, r d n+1 u 



- r ' 



\r 



dv n dv 

 Therefore 



and thus (1) is proved ; we may write (1) for abbreviation thus, 



Put n = 1 in (2) ; then 



d d ) (d d I du du 



as we may write it ; again put n = 2 in (2) ; then 



*.'-*' {d d (d d JeZ d 



Proceeding in this way we obtain 



d d (d d 1 (d d I J d 



EXAMPLES. 



1. Change the independent variable from x to y in the equation 



,d?u du 

 x -j-j + x -j- + u = 0, supposing y = log x. 



Result ^+ U =C. 

 df 



