The collected mathematical papers of Arthur Cayley.

90

[572

572.
THEOREM IN REGARD TO THE HESSIAN OF A QUATERNARY
FUNCTION.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xII. (1873),
pp. 193-197.]
I WISH to put on record the following expression for the Hessian of k + XP'k',
where P, P' are quaternary functions of (x, y, z, w) of the degrees 1, I' respectively,
and X is a constant; the demonstration is tedious enough, but presents no particular
difficulty.
I write (A, B, C, D) for the first derived functions of P; and (a, b, c, d,f, g, h, 1, m, n)
for the second derived functions; and similarly for P'. The Hessian of P is thus
a, h, g,
h, b, f, m
g, f, c, n
1, m, n, d
which is denoted by (abed); moreover, if in this determinant we substitute the accented
letters for the letters of each line successively, the result is denoted by (abed'); and
so if we substitute the accented letters for the letters of each pair of lines successively,
the result is denoted by (abc'd'). Observe that
abed' = (a'8S + b'8b +...) abed and abc'd' = - (a'a, + b'b +...)2 abcd.
The notation (abeD'2) is used to denote the determinant
A', B', C', D',
A', a, h, g, 1
B', b, f, m
C', g, f, c, n
D', 1I,, n, d