14 
ALGEBRAIC INVARIANTS 
Hence 'dF/'dy is identically zero, so that F does not involve 
y explicitly and is a function of g only. 
6. Forms and their Classification. A function like ax 3 +bx 2 y, 
every term of which is of the same total degree in x and y, 
is called homogeneous in x and y. 
A homogeneous rational integral function of x, y, . . . is 
called a form (or quantic) in. x, y, ... . According as the 
number of variables is 1, 2, 3, . . . , or q, the form is called 
unary, binary, ternary, . . . , or q-ary, respectively. Accord 
ing as the form is of the first, second, third, fourth, . . . , or 
pih order in the variables, it is called linear, quadratic, cubic, 
quartic, . , . , or p-ic, respectively. 
For the present we shall deal with binary forms. It is 
found to be advantageous to prefix binomial coefficients to the 
literal coefficients of the form, as in the binary quadratic and 
quartic forms 
ax 2 + 2 bxy-\-cy 2 , aox*+da^y+Qa2X 2 y 2 +dasxy 3 + a±yA. 
7. Definition of Invariants and Covariants of Binary Forms. 
Let the general binary form / of order p, 
aox p +paix p ~ 1 y+^~—~-a2X p ~ 2 y 2 +. . .-\-a v y p , 
1 * ¿j 
be replaced by 
Aoi’ > +pA l e- l n+^^f-A i .e- 2 n 2 +- ■ -+A,^ 
by the transformation T (§5) of determinant A^O. If, for 
every such transformation, a polynomial /(ao, . . . , a v ) has 
the property that 
/(Ao, • • • ? A.f)=A I(ao, . . . , af), 
identically in a 0 , ... , a v , after the A’s have been replaced 
by their values in terms of the a’s, then I{ao, . . . , a p ) is 
called an invariant of index X of the form /.