The theory of determinants in the historical order of development, by Sir Thomas Muir.

HESSIANS (SYLVESTER, 1851)

389

a power of the modulus of transformation." It is spoken of later
in the paper as the "common constant determinant" or the
' ordinary determinant" of the function, the word discriminant
not being proposed until a later date in the same year. In the
second place, there is brought into notice in connection with any
homogeneous integral function p(x, y,..,z) of the nth degree
the family of functions
+... + $, y,...,
ax ay a ''
where r has the values 1, 2,..., n. Corresponding to these
there is a family of determinants (i.e. discriminants), namely
_(X, ~., ),
where r = 2, 3,..., n, the first being according to Sylvester the
"Hessian" or "First Boolian" determinant * of;5, and the last
the "Final Boolian" or "ordinary determinant" of;5. The
reader is left in the former case to reconcile the new definition
with Hesse's own definition, and in the latter case to observe that
( + +... + j(, y ',., ) = ( '* *,.., )
The notation used for the Hessian of s is H(0): by "second
Hessian " he says he means " Hessian of the Hessian "; by "postHessian" the determinant of the function got by taking r=3;
and similarly for "preeter-post-Hessian"!
We may at once remark that much of this nomenclature had
a very short life, being supplanted by other coinages made by
Sylvester himself. The functions
" On p. 194 he says the Hessian of F(x, y) is "the determinant of the determinant, in respect of ~ and V, of
-an e i i atd i t y)
-an error which is repeated in the Collected Mcath. Papers.