82 
Apply S to this term and. the terms resulting until no new terms are obtained, 
then forming the simplest symmetric function of these terms we reach 
Dt rep ele TO 2228 ates oP ae | 
[4 pete Fe +7) y. eter Bera 
This form is invariant under S, but vanishes identically unless a+ 6+ y=o0 
(mod 3). Hence we conclude a + 8 + y=0 (mod. 8)...............000000e (1) 
Apply Tto F,. Immediately there appears, among others, the terms A yA Zp VAP 
and BZ, 223 VA A and B being independent of Z; hence terms of this type are 
in our invariant, and it is necessary to investigate their behavior under the appli- 
cation of S. Treating Ba A Zs exactly as Dias 2 Tig was treated above we 
reach 
Fy =22[ af ay +e? 28 27 4 2h ay|[ise Cty, +y], 
which vanishes identically unless 3+) =o. (mod. 3). 
Similarly treating Z,° zZP Z;, we obtain a form which vanishes identically 
unless a+ 3—o0 (mod 8). Hence we conclude that 
a + 30 (mod 3) 1 ) 
3 + y =(o) (mod 8) J 
but a+ 8+ y=o0 (mod 3). 
-.-a@=o0 (mod 3) 
Z=0 (mod 3) Perr re sree tte eet et etter scenes (3) 
—_o (mod 3) 
It is evident that if t be any three lettered term whose exponents satisfy 
the conditions (3) that T’’: (t) flay a +8 +7 t, 
.'. F,, which is invariant under §, satisfies the relation 
ee Sa rouger rst Soi. ; 
Hence a + 3+ 7-0 (mod 2) is a sufficient and necessary condition that F, is an 
1 
absolute invariant under T?. Similar remarks apply to F, and F,. Hence we 
conclude 
a+ B--y=0 (mod 2) 
.. from 1 
G64 4 =o: (med’6) abyss sheemaeeaees: wnt soar aeeee (4) 
Tf, therefore, Tin ZP VAS is a term of an invariant form, then 
Gap — y—0 (mod 8) |. Me) ms ney Oe (5) 
a+ 3+ y=0 (mod 6) 
