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

380 HISTOIRY OF THE THEOIRY OF DETERMINANTS

En repre'sentant.-par i, j, kc, 1, mn les coefficients diff6rentiels du
quatrie'me ordre de U, on a
I =ikrni - ill - M - 1V + *2jRd,
J = 4jh- - mide manie're que IL J sont des fonctions de x, y des ordres 3(v -4)
et 2 (v-4) respectivement." To this he adds the remark,-able fact,
that if the binary quartic
je + 4jel, ~ 61CC2)12 + 4le,7 + 1)b4
where i, j, 7C, 1, gm are now any quantities independent of $,~be
transformed by the substitution
and I and J thus become I' and J', then
F' = (X~k'-X'k)6I, J' (Xf4'-X',X)4j.
In other words, he makzes known for the first time the two
CCinvariants " of a binary quartic, and notes the curious fact that
expressions of exactly the same form occur in his equivalent for
'V(U2,, + aVU2,4)
Another remark is equally suggestive, namely, that simpler
results might be reached if U were takien a homogeneous function
in x', y' as well as in x, y, and VU were defined as
a2U D2U D2U Da21
'D x/ -y,'ay ay' D 'y x'
For example, U being a quadric in both sets of variables, namely,
U= x2 (AX2 +2Bxy +Cy2)
+ 2x~y1(A 'X2 + 2'B'xy ~ C'y2)
+ y2(A. IX2 + 2B"xy + C1y 2),
or in later notation
U=X2 2xry y2
A B C x12
A' B' C' 2x~y1
A" B" C" Yi