§4] 
EXAMPLES OF INVARIANTS 
11 
F is the square of a linear function of £ and rj, whence the 
discriminant D=AC-B 2 of F is zero. In other words, d = 0 
implies D = 0. By inspection, the coefficient of —b 2 , the highest 
power of h, in the expansion of D is 
(a ô+187) 2 — 4o!7/3 8 = (a ô — /S'y) 2 = A 2 . 
Thus D-A 2 d is a linear function bq+r of b, where q and r are 
functions of a, c, a, p, y, 8^ Let a and_c remain arbitrary, but 
give to b the values Vac and -Vac in turn. Since d = 0 
and D = 0, we have 
0 = Vacq+r, 0 = —Vacq+r, 
whence r = q = 0, D = A 2 d. Thus d is an invariant of / of 
index 2. Another proof is as follows; 
a 7 
a b 
a P 
P 5 
b c 
i 7 s 
a 7 
aa-\-by ap + bô 
A 
B 
P 5 
ba-\~cy bp-\-c8 
B 
C 
We just noted that d = 0 expresses an algebraic property of 
/, that of being a perfect square. To give the related geo 
metrical property, employ homogeneous coordinates for the 
points in a line. Then /=0 represents two points which coin 
cide if and only if d = 0. Thus the vanishing of the invariant 
d of / expresses a projective property of the points represented 
by/=o. 
5. Examples of Covariants. The Hessian (named after 
Otto Hesse) of a function f{x, y) of two variables is defined 
to be 
a 2 / 
a 2 / 
dV 
a*ay 
d 2 f 
a 2 / 
dydx 
ay 2 
Let /become F(%, n) under the transformation 
T : 
x=a£-\-Prj, y — ytj+Srj, 
P 
8 
^0.