178 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
The Jacobian of the two forms, viz. 
2 ax + fiy, fix + 2 7 y 
p , Q 
= x(2aQ-fiP) + y(fiQ-2Py), 
is a linear covariant of the degree 7, say it is 
= P'x + Q'y, 
and it is to be observed that the determinant PQ' — PQ of the two linear forms is 
= - 2 (a, fi, 7- Pf, that is, it is = 2(7. 
310. Hence writing 
T= -1= (P* + Qy) = —X + Y), 
2 Va 2 \/A 
u= -I=(P'*+0'2/) = ^1 ( _ X+ 7), 
whence also 
X = T</A- 
U 
VA' 
U 
Y=T\/A+ 
the determinant of substitution from (X, F) to (T, TJ) is = 2, that from (T, U) to 
(x, y) is 2(7, = and consequently that from (X, F) to (x, y) is =1. 
AT 2 - U*=~ {{fi' 2 - 4a 7 ) (Px + QyY - (P'x + Q'y) 2 }; 
or putting for P', Q' their values, this is ~ 4 (a, fit y^Q> ~ 1 )■ (ax- 4- 2fixy + yy ), 
that is, we have 
A T l - U 2 = ax 2 + fixy + yy-; 
and we have also 
AT 2 - U 2 = i VI [{X + F) 2 - (X - F) 2 ] = \ AX F, 
consequently _ 
ax- + fixy + yy 2 = AT 2 - (7 2 = VAX F. 
311. We have 
Q’T-QU), 
P'T+PU), 
x = -={ 
s/C 
V s/C^