PliOFESSOE A. 11 FCJESYTH OX THE 
2'J8 
flie last Jive involve derivatives of (f) of the second order as ivell as derivatives of the 
fist order, and the first involves derivatives of (f) of the first order alone. 
Every other differential invariant, involving derivatives of ^ of order not higher 
than the second, together with the quantities a, h, c, f g, h and their derivatives of 
the first order, is expressible in terms of the members of this aggregate. Such a 
dilierential invariant is provided by the discriminant of the quantity ©j.,, and its 
expression is 
V = 
51 , 
4 ffb 5 
It is not difficult to prove that 
17. It is also convenient, in view of the expressions for the differential invariants 
of the third order aljout to be considered, to give the umbral symbolical expressions 
for these invariants just considered. We write 
*^100 — 
X = aq. 
^oiu — 
Y = a.,, Z = aq. 
In connection with the first ternary quadratic, we introduce sets of umbral symbols 
cq, cq, (q; hi, b.,, ; and so on. In connection with the second ternary quadratic, 
we introduce sets of umbral symbols o'g; h'l, b'.,, h'g, and so on. Then we 
write 
{a, h, c,f g, hfX, Y, Z)- = of = hf = . . . , 
(a, b, c, f, g, hXX, Y, Z)- = = bfi = . . . ; 
and the seven relative differential invariants are 
A, = ^ {ahcf, 
©1 = 1 {abuf, 
©^o = [aa'idf, 
e, = F {a'Uiif, 
A|.^ = ^{a'ahy, 
— s {na'b)', 
A, = i {a'l/cfi. 
It is known* that another pure contravariant is possessed by two ternary quadratics; 
its symbolical expression is 
(aahi) {ah'c') (a'bc) (hcu) {b'c'u). 
* Clebsch, ‘ Vorlesungcn liber Geometric,’ p. 391. 
