Colloquium publications.

78

THE MADISON COLLOQUIUM.

Hence S is a linear combination of the functions (49) and their
products by b3. Moreover, S must remain unaltered modulo 2
when a3 and bl are replaced by
(50) a3' a3 + al, bl' _ bl + b3 + a2,
which are the only altered coefficients of the form obtained from
F by the transformation
(51) xl - xl', x2 - 2', x xa' + xl' (mod 2).
Both requirements are evidently met by the functions
(52) 1,, 1a2, b3, ala2
and any invariant of F. We find that
A = ala2(a3 + 1), A = la2a3 + j + a3 ab a3,
(53) J = cala2(a3 + 1)(b3 + 1) + b3j + a3b3j + b3q + ala2q,
AJ = 0la2(a3 + 1)(b3 + )(q + 1).
From these and their products by b3, we see that
(54) AJ, baJ, J, bNA, b3A, A, A
contain the respective terms
b3aia2a3q, b3ala2q, ala2q, b3j, b3ala2a3, j, ala2a3,
while no one involves an earlier one of these terms. Hence any
linear combination of the functions (49) and their products by
b3 is a linear combination of the functions (52), (54) and
(55) a3, b3a3, q, b3q, a3q, b3a3q, a3j, b3a3j, ala2a3q.
A linear combination of the latter is of the form
a = mla3 + m2q + m3a3q +- m4a3 j + mala2a3q,
where mi,.*, m4 are linear functions of b3, while m is a constant.
The coefficient of a3bl is seen to be
p = m2 + m3b2 + m4al + mb2al(R + b3 + 1),