236 THE REV. ROBERT HARLEY ON A CERTAIN CLASS 



tions unchanged, we pass at once to the form (E). In 



like manner writing 



bx ;, y 

 — and ^ 

 a™ a 



for w and y respectively, the equation (II) takes the form 

 (IV) ; and making these substitutions in (C), D remain- 

 ing as before unchanged, we obtain the differential re- 

 solvent of (IV), viz. (F). 



The particular cases on which the foregoing inductions 

 were founded are as follows : — 



1. I begin with the form (I), in which I assign to n the 

 successive values 2, 3, 4, 5. There result the equations 



y^ — 2y+ 0^ = 0, 

 y^ — ^y + 2a^ = 0y 

 2/^-42/-}-3<2?=o, 



2/^ -52/ + 4^=0. 



whose resolvents I now proceed to calculate. 



2. In the case of the quadratic, a single differentiation 

 gives 



dy _ I I _ I I 



da^ 2 ' y — i 2 ' 1— <2? 



i^-i), 



or 



the resolvent required. 



3. In the case of the cubic we have, by successive differ- 

 entiations and reductions, 



d^v 



3^(1^^-)- _^=_{3^2^^-|.(i + 2<r^)2/-6<r}. 



Combining these equations so as to eliminate 2/*, and sim- 



