The Interchange of Two Differential Resolvents. 83 



differential resolvent must therefore contain a term indepen- 

 dent of y. This term written on the dexter = x, and the 

 terms on the sinister follow the law indicated in (a'). 

 8. If in equation (a) we write — 



for //, x, y respectively, it becomes — 



3/ -(»' + l)y' +?iV = . . . . ( y ) 

 an equation which is, in form, the same as Q3), and coincides 

 with (/3), when we drop accents and write n— 1 for ;/. Here 

 observe that 



d , , ,x , d 



x 

 or 



^-- (M ' +1 )w 



D= -(»' + l)D' 3 



where, for shortness, D and D' are written for 



d , , d 

 x— and «— -, 

 a« ax 



respectively. 



Effecting in the differential resolvent (a') the same 



substitutions which changed (a) into (y), and reducing by 



means of the formula 



f(D)x r u = x r f(D + r)u, 

 we are led to 



(n + l) n + l [ - ( n - + 1)D' + l]-('"+D 



-n'*[-(n'D' + l)]-^'+»xy = . . . ( 7 ' ) 

 a result which I obtained many years ago, when I was 

 seeking to pass from the differential resolvent of (a) to the 

 differential resolvent of (|3). The form was considered 

 " curious and interesting," and I placed it on record in the 

 Memoirs of this Society, remarking that it involved an 

 " anomaly," and that I should " probably discuss it at some 

 future time." 11 The supposed "anomaly" may be cleared 



""Ona Certain Class, &c." (paper cited in the last footnote) Art. 13, p. 244. 



