234 THE REV. ROBERT HARLEY ON A CERTAIN CLASS 



by {uyj the form (A) may be written 



In the ^ Proceedings' of this Society (vol. ii. pp, 181-184) 

 for the 4th of February last, I gave, without the details of 

 calculation, the several differential resolvents for the suc- 

 cessive cases 9^=2, 3, 4, 5 ; and these results Mr. Rawson 

 of Portsmouth has kindly verified. I gave also in the 

 same paper, the Boolian (symbolical) form of the resolvent 

 for the biquadratic ; and this seems to have suggested to 

 Mr. Cayley an investigation in which he showed, by the 

 aid of Lagrange's theorem, that the equation (B) holds for 

 all values of n. I had the honour of communicating Mr. 

 Cayley's investigation to the Society on the ensuing i8th of 

 February; and an abstract of it appeared at p. 193, vol. ii. 

 of the ^ Proceedings.' The paper itself is printed in this 

 volume of Memoirs, at p. iii. Before receiving Mr. 

 Cayley's remarkable analysis I had calculated, and I 

 beKeve I had also communicated to Mr. Cockle, the 

 Boolian forms of the resolvents for the cases ?^=2 to w=5, 

 both inclusive; and these suggested to me the general 

 form (A). (See ^ Proceedings,^ vol. ii. pp. 199-201, and 

 pp. 237-241.) 



The singular simplicity of these results for the trinomial 

 algebraic equation (I) had an effect in inducing me to con- 

 sider the corresponding form 



y^—ny''-^ + {n—i)a!=o (II) 



to which also any algebraic equation of the nth degree, n 

 being not greater than 5, can, as Mr. Jerrard has shown, 

 be reduced by means of equations of inferior degrees ; and 

 by induction I was led to the following general expression 

 for its resolvent, viz. 



= [n-iy-'e', (C) 



