DIFFEKENTIAL INVARIANTS OF SPACE. 
297 
Now the complete system (being the asyzygetically complete system and not 
merely that which is algebraically complete) of two ternary quadratics is known it 
thus enables us to select the five concomitants required wliich, it is to be rememljered, 
are to involve the quantities a, b, c, f, g, h. Let 
A = be — fh B = ca — C = ab — hq 
F =: g’h — af, G = hf — bg, H = fg — cb, 
Qt ~ be + be — 2/f, 
= ca + ca — 2gg, 
= aha,h— 2h\i, 
5 = gh + gh — rtf — ag; 
@ = Af -f h/ —hg— bg, 
^^=./g'+fi/ — ch —cA: 
and, in order to exhibit the relation of the concomitants to the two quadractic forms, 
let 
U = Aj, e = 0^. 
Then the five quantities required are 
^-b, (5, (5') '^X^iou) 4’wi’ ^ooiY’ 
-B) ^ J G-, 
Ai., = .la + Bh + Cc + 2A’f + 26^g + 2/ih, 
A.^ = Art + BA + Cc + 2Fy’-b 2Gg + 2HA, 
A, = a b g 
b b f 
g f c 
The respective values of g, as determined from the relation 8l + 27n + n = 3g, are 
easily found ; "we have 
Index 4, 0,.,, A^o; 
G, 0^, A^j ; 
8, A.. 
We have already seen that the index of 0j (= 0) is 2 and the index of A^ (= L') is 2. 
Hence an aggregate of algehraicalhj inde[jendent absolute inuariauts, ivhicJt involve 
rt. A, c,f, g, A mid their Jirst derivatives and ■which also involve derivatives of a single 
function cf) up to the second order rnclusive, is composed of 
0 
01-7 
^21 
_»1 
* Clebscu, ‘ Vorlesungeii iiber Geometrie,’ vol. 1, p. 290: the construction of the system is duo to 
Cord AN. 
2 Q 
VOL. CCII.-A. 
