TRANSACTIONS OF SECTION A. 567 



and printed in the second Tolume of tbe third series of the Society's Memoirs, pp. 

 232-245. A few weeks ago on re-studying this result. I succeeded in clearing up 

 the supposed anomaly, and in converting one of the differential resolvents into the 

 other. I will here indicate briefly the method employed, as it appears to admit of 

 general application. 



The differential resolvents of the equations 



• «/"— wy + (n — l).r = . . . • (a) 



2/"-n?/"-i+(n-l).i=0 . , . . O) ■ 



are 



„-.-i^D]"-i-(n-l)« [ " D-?!^^T'"'a;"-iy = . (a') 



Lw — i 71 — i J 



w''-i[(n - 1)D]«- V - (n - l)(JiD - w - 1)[«D - 2y'-^xi/ = [n- Vf'-Kv . (3') 

 respectively, where D =-^'^~, and the usual factorial notation 



[^]«=(^)(^_1)(^_2) . . . ((9-a+l) 



is followed. The question is, how to pass from (a') to {8') ; or, in other words, 

 given the differential resolvent of (a) to find that of (/3). The following method 

 is effective. 



If in equation (a) we write 



, /■n'"'-^\ 1- / 1 x_i_ 

 '\x' ) ' Vn'VJ y 

 for n, X, y respectively, it becomes 



y'+i-(?i' + l)/"' + w'.i-' = . . . . (y) 

 which is of the same form as (/3). Here 



.r 'L = - (,i' + Y)x'—, or D = - («' + 1)D'. 



These substitutions being made in the resolvent (a') we are led to 



(rt' + l)"'+i[ - (n' + 1)D' + l]-"''+i)^ - n'\ - (n'D + l)]-"*'+i'.i>' = . (y') 



the result given in the paper above cited. 

 Now, observing that, in general, 



_ (-1)1 



L-< 



\_e-\Y 



we have 



and 



80 that (7') may be written in the form 



(«' + l)"'+i[«'D']»'+iy'-n'=[(w' + l)D'-2]'''+ia-y = , . (y'j) 



which contains the common factor (D — 1), and therefore admits of a first integra- 

 tion, Operating with (D — 1)"^ , and determining the constant by summing for 

 the (n+1) roots of i/'{2y' = n' +1), we obtain the differential resolvent of (y), 

 namely, 



(n' + l)»'[n'D']''V' - n\7i' + ID' - n^T2)[(«' + 1)D' - 2]"'- Vy' = [w']'''.f' . (y',) 



Dropping accents and writing n — 1 for n, we are conducted finally to the equation 

 0'), the differential resolvent of (/3). 



