83 
: ‘ EO aides ; : 
Applying now S and T to Z, Z) Z; and to the terms resulting until no 
new ones are reached, and forming the simplest symmetric functions, we obtain 
the form: 
Boy Briers y | 
7 ASD NOL lay AY ela ea RAY Ad 
é f ¥, ig 3 7 7 ig 5 7) 
| RIGA aL I al Aaah ay Sele Aa 9g) 
6 Dwy ig ira ig BU ay 
eh Aware ae Zr Bl One Ae | 
PRET zea! n+ Ad oe af 22 ZF at \=Pa, B, >, (aay), (0): 
Giving to a, 3, y all possible values under limitations (5), we obtain a triply 
infinite system of invariant forms. 
And this is the complete system, for we cav only start with 
7 an Vga t op PS OMea ae oe 6) ) if, Dg.) 
2 OP De, Oe OP 22, 22 OP a or BE ZEAL. 
But all these terms may be found in | oe 3) by a suitable interchange of «, }3, y. 
A 
3. » is also an invariant. Since 
Certainly any rational function of the forms ps ; 
Dt fad, 
a+ 3+ y=o (2), at least one of the exponents is even. 
Corollary]. Py ga g?7h 472 7P 4 72 7h 4 oooh + 2e oP + ae ae 
Pe eee A ae Boe Me ere Le eee Ze De \entes 
Cor, 2. Po. pe = Pe, a; Po ee 
Cor. 3. If 6 =o and y—o, and a = 0 (mod 2), we get a one-lettered form, 
Pi bat (t=1 == 4). 
a= o(mod 6). 
2. To calculate the basis of our system. 
*Hilbert has shown that if Z isa system of integral homogeneous forms of 
n variables that F, any form of Z can be represented by the expression 
mm 
F=Z A; F; when m is finite and A; are homogeneous integral functions of the 
i=1 
variables and F; are forms of Z. 
And in particular if Z be defined as a system of forms invariant under a group, 
that F any form of Z may be represented by a rational integral function of a 
finite number of forms of % This finite number of forms is called the basis of 
the system. 
*See Mathematische Annallen, Vol. 36. 
