405] 
AX EIGHTH MEMOIR ON QUANTICS. 
179 
so that, pausing a moment to consider the transformation from (x, y) to (T, U), we 
have 
(a, b, c, d, e,f\x, y)*=~~(a, b, c, d, e,f^Q'T-QU, -P'T+PUy 
= ^ 5 (a, b, c, d, e, f][T, U) 5 suppose, 
where (a, b, c, d, e, f) are invariants, ot the degrees 36, 34, 32, 30, 28, 26 respectively 
it follows that b, d, f each of them contain as a factor the 18-thic 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 
{y^x} = cid b + 2bd c + 3cd d + 4>dd e + oedf, = 8 suppose, 
[mdy] = obd a + 4 cd b + ¿dd c + 2 ed^ + fd c , = 8 } „ 
then Px + Qy, Px + Q'y being covariants, we have 8P =0, 8Q = P, 8P' = 0, 8Q' = P', 
whence, treating T, U as constants, 8 {Q'T - QU) = P'T — PU, 8 (— P'T + PU) =0. Hence 
8(a, b, c, d, e,/WT-QU, -P'T+PU) 5 
= 5 (a, b, c, d, e\Q'T-QU, - P'T + PUf.(- P'T + PU) 
+ 5 (a,b,c,d,el „ „ )*.( P'T - PU) 
+ 5 (b, c, d, e, /5 „ „ ) 4 . 0, 
the three lines arising from the operation with 8 on the coefficients (a, b, c, d, e, f) 
and on the facients Q'T—QU and -P'T+PU respectively; the third line vanishes of 
itself, and the other two destroy each other, that is, 
8 (a, b, c, d, e, f\Q'T- QU, -P'T+PUf = 0, and similarly 
8, (a, b, c, d, e,flQ'T-QU, -P’T+PUy = 0, 
or the function (a, b, c, d, e,f\Q'T-QU, -PT + PUy, treating therein T and U as 
constants, is an invariant, that is, the coefficients of the several teims theieof are all 
invariants. 
313. The expressions for the coefficients (a, b, c, d, e, f) are in the first instance 
obtained in the forms 
a = 2 (L + oMG+lOG 3 ), 
b = — 2 (2/ + SM'G) A, 
c = 2 (L + MC — 2C-) 
d = -2 (L'-M'G), 
e = 2 (L - SMC + 20 2 ) A~\ 
f = — 2 (/>' — oM'C) A~ l , 
23—2