NOTES AND REFERENCES. 
601 
true for all values of e, e. Again suppose that G alters neither when x receives such 
infinitesimal increment, y and 0 remaining constant, nor when y and z separately 
receive corresponding increments z, x and x, y in the respective cases remaining 
constant. ...G will remain constant for any concurrent linear transformations of x, y, z 
when the modulus is unity. This all-important principle...also instantaneously gives 
the necessary and sufficient conditions to which an invariant of any given order of 
any homogeneous function whatever is subject, and thereby reduces the problem of 
discovering invariants to a definite form.” And in section 6 of the same paper 
(Sylvester 11) referring to the Note, he writes “This method may also be extended 
to concomitants generally. M. Aronhold as I collect from private information was the 
first to think of the application of this method to the subject: but it was Mr Cayley 
who communicated to me the equations which define the invariants of functions of 
two variables. The method by which I obtain these equations and prove their 
sufficiency is my own, but I believe has been adopted by Mr Cayley in a Memoir 
about to be published in Grelle’s Journal [? 100]. I have also recently been informed 
of a paper about to appear in Liouville’s Journal from the pen of M. Eisenstein, where 
it appears that the same idea and mode of treatment have been made use of. 
Mr Cayley’s communication to me was made in the early part of December last 
[1851] and my method (the result of a remark made long before) of obtaining these 
and the more general equations and of demonstrating their sufficiency imparted a few 
weeks subsequently—I believe between January and February of the present year 
[1852],” and then applying the principle to the binary quadric, he proceeds to consider 
d d 
26^ + 3c^ + ..., and the other operator with the 
coefficients in the reverse order, as applied to an invariant <£ of the quantic. The 
theory of these operators was thus familiar to Sylvester in 1852, but it was in 
nowise made the foundation of the structure. 
I notice as contained in the paper Boole (4), what is probably the first state 
ment of the “ provectant ” process of forming an invariant; for example, from the 
quartic function (a, b, c, d, e\x, y) 4 he derives 
Jg (a, b, c, d, etfdy, — d x y . (a, b, c, d, e$x, y) 4 = ae — 46d + 3c 2 , the quadrinvariant; 
and similarly from the Hessian (ac — b 2 , 2 (ad —be), ae + 2bd — Sc 2 ) 2 (be — cd), ce — d^x, y) 4 
is derived the cubinvariant ace — ad 2 — b 2 e + 2bed — 3c 2 . Mention is also made of the 
function A (/38 - y 2 ) + B (/3y -a 8) + G (ay - /3 2 ), (A, B, G given quadric functions, a, /3, y, 8 
given cubic functions of (a, b, c, d, e, /)), which is the octinvariant Q of the binary 
quintic. 
The papers of Sylvester contain a great number of important results which will 
some of them be referred to in connexion with the later Memoirs on Quantics. 
Hermite’s discovery of the invariant of the degree 18 of the quintic, and the 
demonstration of his law of reciprocity are both given in the Memoir by him which 
is above referred to. 
d 
the theory of the operator a ^ + 
a il 
76