I 



OF LINEAR DIFFERENTIAL EQUATIONS. 243 



I have also verified this resolvent by means of other 

 numerical values of y. These verifications seem to place 

 the accuracy of the sextic resolvent beyond doubt^ and to 

 afibrd additional confirmation of the generality of the 

 form (C) or (D). 



13. There is probably some method of passing directly 

 from the diflPerential resolvent (A) to the resolvent (C). 

 The algebraic equations (I) and (II) _, from which they are 

 derived_, are closely related, and may easily be deduced 

 the one from the other. 



1°. If in equation (I) we write 



for X, y respectively, it becomes 



y'^—ny''^-'^ + {n—i)x=Oj 



an equation which, the accents being suppressed, is iden- 

 tical with (II) . 



In this transformation it is to be observed that 



d , . , d 

 or 



2°. Or, if in equation (I) we write 



for Thy X, y respectively, it becomes 



?/' —\rv-\-\)y ■j-nx=o, 



* More generally: If x=^x', then we have 



d . d .„ \ d 

 a?_ =^w — =-6u . — . — 



dx d<pu (p'u du 



(where the accent denotes differentiation), 



whence, if x_=w, or ^ — =-, and <pu=u*', then will 



d -id 

 ax du 



r2 



