584 
NOTES AND REFERENCES. 
8. A correction to the theorem (18), p. 42, is made in my paper “Notes on 
Lagrange’s theorem,” Carnb. and Buhl. Math. Jour. vol. vi. (1851), pp. 37—45. 
10. This paper is connected with 5, but it is a particular investigation to which 
I attach little value. The like remark applies to 40. 
12. The second part of this paper, pp. 75—80, relates to the functions obtained 
from n columns of symbolical numbers in such manner as a determinant is obtained 
from 2 columns, and which are consequently sums of determinants: they are the 
functions which have since been called Commutants; the term is due to Sylvester. 
13. In modern language: Boole (in his paper “ Exposition of a general theory 
of linear transformations,” Carnb. Math. Jour. vol. in. (1843), pp. 1—20 and 106—119) 
had previously shown that a discriminant was an invariant; and Hesse in the paper 
“Ueber die Wendepunkte der Curven dritter Ordnung,” Crelle, t. xxvm. (1844), pp. 68— 
96, had established certain covariantive properties of the ternary cubic function. I first 
proposed in this paper the general problem of invariants (that is, functions of the 
coefficients, invariantive for a linear transformation of the facients), treating it by what 
may be called the “ tantipartite ” theory: the idea is best seen from the example p. 89, 
viz. for the tripartite function 
TJ = ax 1 y 1 z l + bx 2 y x z x + cx x y 2 z x + dx 2 y 2 z x + ex x y x z 2 + fx 2 y x z 2 + gx x y 2 z 2 + hx 2 y 2 z 2 , 
we have a function of the coefficients which is simultaneously of the forms 
H 
a, b, c, 
d 
, H 
a, b, e, f 
, H 
Qjy C j ß) 
9 
e > f 9’ 
h 
c, d, g, h 
b, d, f 
h 
and as such it is invariantive for linear transformations of the (x x , x 2 ), (y x , y 2 ), (z x , z 2 ). 
Passing from the tantipartite form to a binary form, I obtained for the binary 
quartic the quadrinvariant (I = ) ae — 4bd + 3c 2 : as noticed at the end of the paper, 
the remark that there is also the cubinvariant (J = ) ace — ad? — b 2 e — c 3 + 2bed was 
due to Boole. The two functions present themselves, but without reference to the 
invariantive property and not in an explicit form, in Cauchy’s Memoir “ Sur la déter 
mination du nombre des racines réelles dans les équations algébriques,” Jour. École 
Polyt. t. x. (1815), pp. 457—548. 
In p. 92 it is assumed that the invariant called du is the discriminant of the 
function U = ax x y x z x w x ... + / px 2 y 2 z 2 w 2 : but, as mentioned in [ ], the assumption was 
incorrect. This was shown by Schlafli in his Memoir, “Ueber die Résultante eines 
Systèmes mehrerer algebraischen Gleichungen,” Wiener Denies, t. IY. Abth. 2 (1852), 
pp. 1—74: see pp. 35 et seq. The discriminant is there found by actual calculation to 
be a function (not of the order 6 as is 6u, but) of the order 24, not breaking up 
into factors; in the particular case where the coefficients a,...p are equal, 1, 4, 6, 4, 1 
of them to a, b, c, d, e respectively, in such wise that changing only the variables 
the function becomes = (a, b, c, d, eQx, y) 4 , then the discriminant in question does break