this gives 
and thence 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
545 
ad®— ffofi w +ybl=— v'Adxdy, 
(ad®— /3d,d,+ yd2)®(a, b , c, d, e,f\x, yj 
=Ad|d®(X, p, v', (a x'XX, Y) 6 
=120A(^X+/Y); 
the left hand side is a linear covariant of the degree 5, it is consequently a mere nume- 
rical multiple of P#+Q y; and it is easy to verify that it is =120(P^+Qy). (In fact 
writing b=d=e= 0, the expression is ( 3c®d® — afd^> x f{ax b + 10ar 3 ^® -\-fy 5 ), and the only 
term which contains x is a?f* . d®d® . 10c# 3 ;y®=120«®c/ > ® . x; but for b=d~e— 0, Table 
No. 22 gives P x=a 2 cf 2 x, and the coefficient 120 is thus verified.) But P^-f-Qy is 
=-^=(X-(-Y), and we have thus Av=A whence not only v=v', = v'k suppose, 
y A y A 
C 
but we have further Jc= - 77 =’ a result given by M. Hermite. 
315. Substituting for v=v' the value \/ k , we have 
(a, b , c, d, e,fjx, yf 
=(«, i, « 4. t,/X ^(|j+Q X +M Ia- q **)y, 
=(X, ft, -s /&, s/k, \d, x'JX, Y) 5 , 
and we have then ax 2 + fixy + yy <i= s/ AXY, viz. the left-hand side being the quadrico- 
variant of (a, b, c, d, e , fX.x, yf, the equation shows that the quadricovariant of the form 
(X, [a , \/ k, \/k (d, X'XX, Y) 5 is =\/ AXY, and we thus arrive at the starting-point of 
Hermite’s theory. 
316. The coefficients (X, p, *Jk , \/k, (d, X') of Hermite’s form are by what precedes 
invariants ; they are consequently expressible in terms of the invariants A, B, C (and I). 
M. Hermite writes 
and he finds 
XX' =< 7 , \jj\d-=.h , 
*/A.=g-3h+U, 
A 6 
=k. 
or, what is the same thing, 
A S + 3AB + C 
h= 
AB + C 
" Va 5 7 v"a 5 
which give g , h, k in terms of A, B, C, and then putting 
7 c 
k— 
v A 5 
I 2 
A=(9£®+16M-^)®-24M 3 , =^ 7 
(the equation I®=A 7 A is in fact equivalent to the before-mentioned expression of I® 
mdccclxvii. 4 E 
