PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
543 
or putting for F, Q' their values, this is into 4(a, (3, y^Q, — P) 2 (a# 2 + 2flxy + yy' 1 ). 
that is, we have 
AT 2 — U 2 = a# 2 + /3 xy + yy 2 ; 
and we have also 
AT 2 - U 2 =|VA[(X+ Y) 2 - (X- Y) 2 ] - yAXY, 
consequently 
ax 2 + j3 xy -f- yy 2 = AT 2 — U 2 = \/ AX Y. 
311. We have 
*=^g( Q'T-QU), 
y=^e(-PT+PU), 
so that, pausing a moment to consider the transformation from (x,y) to (T, U), we have 
(a, i, c, d, e,fXx, yf= +;(«, i, c, d, e,/XQ l T-QU, -P'T+PUf 
= -Ayfa, b, c, d, e, fXT, U) 5 suppose, 
where (a, b, c, d, e, f) are invariants , of the degrees 36, 34, 32, 30, 28, 26 respectively; 
it follows that b, d, f each of them contain as a factor the 18thic invariant I, the 
remaining factors being of the orders 16, 12, 8 respectively. 
312. That (a, b, c, d, e, f) are invariants is almost self-evident; it may however be 
demonstrated as follows. Writing 
{yh x }=ad b + 25<++3<?c) rf + 4eZd e +5ed^=& suppose, 
{^}=55d a +4cd i +3dd c +2ed d +/d e , =&, „ 
then P.r+ Qy, P'#+Q 'y being covariants, we have £P = 0, SQ=P, SP'=0, SQ'=P', 
whence, treating T, U as constants, S(Q'T— QU)=P'T— PU, S( — P'T+PU) = 0. Hence 
&(®, b, c, d, e,fX Q'T-QU, -P'T+PU) 5 
=5 (a, b, c, d, ^X^'T-QU, -P'T+PU) 4 . (-P'T+PU) 
+5 (a,b,c,d,eX „ „ ) 4 . ( P'T-PU) 
+5 (b,c,d,e,fX j? „ Y -0, 
the three lines arising from the operation with h on the coefficients (a, b, c, d, e, f) and 
on thefacients Q'T— QU and —P'T+PU respectively; the third line vanishes of itself, 
and the other two destroy each other, that is, 
l (a, b , c, d, e,fX Q'T— QU, -P'T+PU) 5 = 0, and similarly 
\(a, b, c, d, e,fX Q'T-QU, -P'T+PD) 5 =0, 
or the function (a, b, c , d , e, fX Q'T— QU, — PT+PU) 5 , treating therein T and U as 
constants, is an invariant, that is, the coefficients of the several terms thereof are all inva- 
riants. 
