The IntcrcJiange of Tzvo Differential Resolvents. 85 



or, more simply, 



(ri + l)'" +1 [VD']"'+y - ri\(n' + 1 )D' - 2] ,i '+Vy' = 0, 

 an equation which contains the factor (D'-i). The in- 

 tegration gives 



(ri + l) n,+1 [>'D'] n y - ri(ri + ID' - ri + 2) 



[(n' + l)Ty-2] ll '- l x'y' = Cx, 



an equation which must be satisfied by any one of the 

 roots of (7). To determine the constant, sum for the 

 («'+ 1) roots ; then, since Sy = ;/'+ 1, we have 



(ri + 1)" + 1 [riD'] n '(ri + 1 ) - n'(n + ID' - ri + 2) 



[(ri + 1)D' - 2f'- 1 x (ri + 1) = C(»' + l)x, 



which, reducing by the aid of the formula 



/(D>>=a;'/(r), 

 gives 



C = [rif. 



Hence the differential resolvent of (7) is 



(ri + \) n '[riT>'Y'y - n'(ri + ID' - w' - 2) 



[(»' + l)D'-2]"'-V3/' = [ra'] ,l V . . . (y). 



Drop the accents and write n~ 1 for n; the result is the 

 differential resolvent of (j3), viz. : 



n n -\(n - 1)D] B "V - (» - 1)(mD - w - 1)[>D - 2]"-% 



= [w. — lj" _1 x, 



an equation which coincides in every point with Q3'). 

 9. It may be noticed here that if we write 



-<»'-i>. (^n- 



((w'-l)VW' 



for n, x, y respectively, and after substitution, drop the 

 accents, (a) will be changed into (j3) ; also that the same 

 substitutions enable us, by the foregoing process, to change 

 directly (a) into Q3'). 



10. 1 now proceed to show how a certain set of sub- 

 stitutions which change (j3) into (a), will also change (j3') 



