542 
PROFESSOR CAYLEY’S EIGHTH MEMOIE ON QUANTICS. 
309. Reversing the order of the several steps, the theory of M. Hermite’s transforma- 
tion may be established as follows : — 
Starting from the quintic 
(a, b, c, d, e,fjx, y)\ 
and considering the quadricovariant thereof 
(a, /3, y^yz, y ) 2 Tab. No. 14 
((a, /3, y) are of the degree 2), and also the linear covariant 
P.z+Qy . Tab. No. 22 
((P, Q) are of the degree 5), we have 
/3 2 — 4ay=A, Tab. No. 19, 
and moreover 
(«, /3,7l Q, -P) 2 =-C, 
viz. the expression on the left hand, which is of the degree 12, and which is obviously 
an invariant, is = — C, where C is (ut sujprd) 
C=9L+JK= — 9 No. 29+(No. 19)(No. 25). 
The Jacobian of the two forms, viz. 
2«a?+/3 y, fix+tyy , 
P , Q 
=*(2 a Q-j3P)+y(/3Q-2Py), 
is a linear covariant of the degree 7, say it is 
=Far+Q'y, 
and it' is to be observed that the determinant PQ'— P'Q of the two linear forms is 
= — 2 (a, /3, y%Q, — P) 2 , that is, it is =2C. 
310. Hence writing 
T =^( p *+ Q ^=^( X + Y )’ 
u =J7c (p '* +Q W =J ir (- X + Y )’ 
whence also 
Y =T^+-^ 
the determinant of substitution from (X, Y) to (T, U) is =2, that from (T, U) to (x,y) 
is -£T 2C, =^, and consequently that from (X, Y) to (. v , y ) is =1. 
We have 
Ar-U==i|(/3»-4 ay )(P S :+Qy) 5 -(P'*+Q'j,)j ; 
