NOTES AND REFERENCES. 
585 
up into factors, the value in this case being J G (I 3 — 27J 2 ) of the order 24 as in the 
general case, but containing the factor / 3 — 27J~ which is the discriminant of the binary 
quartic. 
14. In this paper I developed what (to give it a distinctive name) may be called 
the “ hyperdeterminant ” theory, viz. the expressions considered are of the form 
where after the differentiations the variables (oc ly y^}, {x 2 , y 2 ),... are to be or may be 
put equal to each other: it is to be noticed that although in the examples I chiefly 
consider constant derivatives, or invariants, the memoir throughout relates as well to 
covariants as invariants. The theory is to be distinguished from Gordan’s process of 
Ueberschiebung, or derivational theory, viz. this may be considered as dealing exclusively, 
or nearly so, with the single class of derivatives (F, TF) a , = 12 a V 1 W 2 : the theorem that 
all the covariants of a binary function can be obtained successively by operating in 
this manner on the function itself and a covariant of the next inferior degree was a very 
important one. 
15. Eisenstein’s theorem may be stated as follows : the function a 2 d 2 + 4ac 3 — 6abcd 
+ 4b 3 d — 3b 2 c 2 (which is the discriminant of the binary cubic (a, b, c, d\x, y) 3 ) is 
automorphic, viz. it is converted into a power of itself when for a, b, c, d we substitute 
dcf> dcf) dcf> d(f> 
the differential coefficients 
~ of the function itself. It is remarkable, 
da ’ db ’ do ’ dd 
see 54, that the function is automorphic in a different manner, viz. the Hessian 
a power of the function itself. The first part of the paper relates to the function 
a 2 h 2 4- b 2 g 2 + ... + 4tbceh which had presented itself to me, 13, in the theory of linear 
transformations, and which is in like manner automorphic for the change a, b, ... into 
of the composition of quadratic forms, Gauss, Disquisitiones Arithmetical (1801), and 
see 92. The second part gives for the binary quartic covariant an automorphic 
formula analogous to those previously obtained by Hesse for the ternary cubic, viz. 
the Hessian of any linear function of the quartic and its Hessian, is itself a linear 
function of the quartic and its Hessian, the coefficients depending on the invariants 
I, J of the quartic form. 
16. This is a mere reproduction of 13 and 14, and requires no remark. 
19 and 23. These papers contain a mere sketch of the application of the doubly 
infinite product expression of the elliptic function sn u to the problem of transformation. 
As noticed in 23, I purposely abstained from any consideration of the infinite limiting 
values of m and n. 
C. 
74