$2 The Rev. Robert Harley on 



(to which all algebraic equations of a degree not greater 

 than 5 may be reduced,) are 



•■"i-sj y - ( *- i) l^i^-^tj y • ■ (a> 



^{n-\)J d ^'l-{n-l)[n4 x -n-l) 



respectively, where the usual factorial notation 



[a] b = (a)(a-l)(a-2) . . . (a- 6+1) 



is adopted. 



7. I may observe here that these forms were obtained 

 originally by induction. 10 The determination of the 

 differential resolvents of the two trinomial equations (a) and 

 C/3) for the particular cases n = 2, 3, 4, 5, on which the 

 induction was founded, necessitated many complicated and 

 laborious calculations, which, however, led in all cases to 

 remarkably simple and uniform results. Much of the 

 labour might have been saved had I noticed at the time, 

 what I now proceed to show, viz., that either of the general 

 forms is implicitly contained in the other ; in fact, I might 

 have derived (/3') from (a'), or vice versa, merely by a 

 change of the variables. When once the general forms had 

 been suggested, there was not much difficulty in completing 

 the induction, and showing that the equations held for all 

 values of n, excepting only n = 2. The exception occurs in 

 the first form (a), which, when n = 2, should evidently coin- 

 cide with (j3')i seeing that in this case, (a) and (/3) become 

 identical. Now in (a) when n = 2, the sum of the roots 

 (Sj) is not, as in other cases, equal to zero, and the 



10 "On a Certain Class of Linear Differential Equations." Manchester 

 Memoirs, Vol. II., Third Series, 1861-62, pp. 232-245. 



" On the Theory of the Transcendental Solution of Algebraic Equations." 

 Quarterly Journal of Mathematics, 1862, Vol. V., pp. 337-360. 



