Colloquium publications.

32

THE MADISON COLLOQUIUM.

we may employ the functions (46) to delete from (44)
2 Ao, A0 o2, aA022,, 2, X2, 1
in turn (no one of these terms being reintroduced at a later stage).
There remain 11 seminvariants of odd weight
(48) AoiA1, AoiA1a, AoiA1a2, a, Ao2A (i = 0, 1, 2),
and 15 of even weight
(49) Ao, Ao2, AoA12, Aoo, Aoa2, Ao2a, Ao2a2, AoiX, A0X2, A02X2, Ao 2.
Now the weight and index of a seminvariant of Fs modulo 3
are both even or both odd.* A linear combination of the functions (48) which is changed in sign by the substitution (AoA3)
(A1A2) is seen to be identically zero (it suffices to set As = 0,
A2 = 0 in turn). A linear combination of the functions (49)
which is unaltered by that substitution is seen similarly to be
identically zero. Hencet a complete set of linearly independent
invariants of F3 modulo 3 is given by (46).
* When the sign of y is changed, a seminvariant is unaltered or changed in
sign according as its weight is even or odd.
t Another proof, using the classes of F3 under the group of all binary linear
transformations of determinant unity modulo 3, and involving a use of more
technical theory of numbers, is given in Transactions of the American Mathematical Society, vol. 10 (1909), pp. 149-154. The case of any modulus p
is there treated.