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1. Mr. Harley has shown that any root of the equationy n —ny +(n—1x =0 satisfies the differential equationy ‒ (D‒2n ‒1/n ) (D‒3n ‒2/n ) . . (D‒n 2 ‒n + 1/n )/D(D‒1) . . (D‒n + 1)e (n ‒1)θ y =0, . . . (1) in whiche θ =x , and D=d /dθ provided thatn be a positive integer greater than 2. This result, demonstrated for particular values of , and raised by induction into a general theorem, was subsequently established rigorously by Mr. Cayley by means of Lagrange’s theorem. For the case ofn =2, the differential equation was found by Mr. Harley to bey ‒D‒3/2/De θ y =1/2e θ (2) Solving these differential equations for the particular cases ofn =2 andn =3, Mr. Harley arrived at the actual expression of the roots of the given algebraic equation for these cases. That all algebraic equations up to the fifth degree can be reduced to the above trinomial form, is well known.

XIX. On the differential equations which determine the form of the roots of algebraic equations