600 
NOTES AND REFERENCES. 
12. Note on the Calculus of Forms, Camb. and Dnbl. Math. Jour. t. vm. 1853 
pp. 62—64 
13. On the Calculus of Forms otherwise the theory of Invariants. Sect. 7. On 
Combinants, Camb. and Dubl. Math. Jour. t. vm. 1853, pp. 256—269. 
14. On the Calculus of Forms otherwise the theory of Invariants. Sect. 7. Con 
tinued. 8. On the reduction of a sextic function of two variables to its canonical 
form, Camb. and Dubl. Math. Jour. t. ix. 1854, pp. 85, 103. 
Salmon. Exercises in the Hyperdeterminant Calculus, Camb. and Dubl. Math. Jour. 
t. ix. 1854, pp. 19—33. 
Hermite. Sur la théorie des fonctions homogènes à deux indéterminées, Camb. 
and Dubl. Math. Jour. t. ix. 1854, pp. 172—217. 
Aronhold. Zur Théorie der homogenen Functionen von drei Variabeln, Crelle 
t. xxxix. 1850, pp. 140—159. 
In the present Memoir 139, dropping altogether the consideration of linear trans 
formations, I start from the notion of certain operations upon the constants and 
facients of a quantic, viz. if to fix the ideas we consider the case of a binary 
quantic (a, b,...b', a'fix, y) m , then there is an operation {yd x }, = ad b + 2bd c ...+mb'd a ' 
which performed upon the quantic is tantamount to the operation yd x : and similarly 
an operation [xd y ], = mbd a +(m— l)cd b ... + a'dy which performed upon the quantic is 
tantamount to the operation xd y . Or, what is the same thing, there are two opera 
tions [yd^ — yd x , and [xd y ) — xd y each of which performed upon the quantic reduces 
it to zero : to use an expression subsequently introduced, say each of these is an 
annihilator of the quantic. The assumed definition is that any function of the 
coefficients and variables which is reduced to zero by each of these operators, is a 
Covariant : and in particular if the function contain the coefficients only (in which 
case obviously the operators may be reduced to {yd x } and {xd y } respectively) the 
function is an Invariant. 
I believe I actually arrived at the notion by the simple remark, say that ad b + 2bd c 
operating upon ac — b 2 reduced it to zero, and that the same operation performed 
upon ax 2 + 2bxy + cy 2 reduced it to 2axy + 2by 2 which is = yd x {ax 2 + 2bxy +.cy 2 }. But 
the earliest published mention of the notion is in the year 1852 in Note 7 of 
Sylvester’s paper on the Principles of the Calculus of Forms (Sylvester 10). Here, 
connecting it with the theory of linear transformations, he writes “There is one 
principle of paramount importance which has not been touched upon in the preceding- 
pages,... The principle now in question consists in introducing the idea of continuous 
or infinitesimal variation into the theory. To fix the ideas suppose G to be a 
function of the coefficients of <£ (x, y, z) such that it remains unaltered when x, y, z 
become respectively fix, gy, hz, where fgh = 1. Next suppose that C does not alter 
when x becomes x + ey + ez, where e, e are indefinitely small ; it is easily and 
obviously demonstrable that if this be true for e, e indefinitely small, it must be