Abstract
1. Mr. Harley has shown that any root of the equation
y
n
—
ny
+(n—1
x
=0 satisfies the differential equation
y
‒ (D‒2
n
‒1/
n
) (D‒3
n
‒2/
n
) . . (D‒
n
2
‒
n
+ 1/
n
)/D(D‒1) . . (D‒
n
+ 1)
e
(
n
‒1)
θ
y
=0, . . . (1) in which
e
θ
=
x
, and D=
d
/
dθ
provided that
n
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 of
n
=2, the differential equation was found by Mr. Harley to be
y
‒D‒3/2/D
e
θ
y
=1/2
e
θ
............(2) Solving these differential equations for the particular cases of
n
=2 and
n
=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.
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