DIFFERENTIAL INVARIANTS OF SI'ACE. 
303 
quadratic as A.,,, A.^, ©o to the second quadratic. Moreover, in the case of 
two surfaces <j) and (f)', the aggregate of invariants of the tirst order is 
Aj, ©1, D©i, 1)'©, ; 
and these must be included in the desired aggregate. Further, as regards the 
concomitants, the second and the third quadratics will have invariants, standing to them 
in the same relations as A^^ and A.,^ to the first and the second or as A^g and Ag^ to 
the first and the tliird : we therefore ha^"e A.,., and A.,., as invariants. 
Accordingly, we take the algebraic aggregate of sixteen relative invariants as 
composed of tlie ([uantities 
© 
^12’ 
■^20 ^2’ 
^'"h2» 
-^13) 
<I 
^">13, 
<'“>3> 
D©1, D'^©g, 
‘^23) 
'^32- 
Let 2Q denote the 
operator 
4wo 
0 
100 
a 
+ 4^001 
0 
^4* 010 
^4^'001 
invariants are given 
by 
then other relative 
D©.,, It©!-., L)'0io, F)©^g, DdHjjg, 
(I©3, (IhHj 
3’ 
and so on; all of these are algebraically expressible in terms of the above aggregate 
of sixteen. 
The umbral expressions for these invariants are easily obtained. We take the 
notation of § 17, and we furtlier introduce a third set of umbral symbols a'\, ; 
h'\, h"n, b ".^; and so on, defined as 
We also write 
(a', b', c', f, g-', h'lX, Y, Zf = aV = b"/ = . . 
4 ^ lUU — n 4 ^ UlO — " 2’ 4 * OUl — 3 ’ 
and then we have the following expressions for tlie sixteen relative invariants :— 
i {abcf, 
^12 — 2 {^-<''^by, 
Ajg = ^ {a"aby, 
i {awy\ 
= h {(iCi'b'y, 
A, 3 1 {a'Wb'f, 
i {a"b"ey, 
A 31 = i {aa"b"y\ 
A,, = ^{a'a%"y\ 
i b^fbuf, 
®i 2 = {aa'uf, 
i {o'b'uf, 
i {a"b"a'f, 
©13 = (rta'V)^ 
II 
[aim) [abu'), D’©^ = 
