Colloquium publications.

INVARIANTS AND NUMBER THEORY.

79

where R b3 + a2 is the increment to b1 in (50). Set
(56) a=pa3bl+ra3+sbl+t (p,. t, t independent of a3, b1).
Let the substitution (50) replace a by a'. Then
(57) a' - a = pRa3 + palbl + palR + ra1 + sR.
This is zero for every a3, b1 if and only if
(58) pR 0, pal 0, ra1 a sR (mod 2).
For p = m2 + **, pal- 0 gives m3 -0, m4 m2. Then
pR 0 gives m2 E 0, mb3 0, whence m -0. Thus a = mla3,
so that mi _ 0. Hence the leader of a covariant of F has the form
(59) I + b31 + cacia2 + dala2bs,
where I and I1 are invariants, c and d are constants.
COVARIANTS WHOSE LEADERS ARE NOT ZERO, ~~ 11-19
11. Consider a covariant of odd order w:
(60) C = SX3a + S1X30'-xl + S2X3W2xL2 + *
If S1' is derived from S1 by the substitution (50), then, by (51),
(61) Si' Si + oS Si + S (mod 2).
Give S1 the notation (56). Then S is given by (57) and has no
term with the factor a3b1. Now a3b1 enters no term of (59)
except J and AJ of I and* b3J of b3I1, and in these is multiplied by
(62) b3al + ala2, 0aa2(b2 + 1)(b3 + 1), b3a1a2,
respectively. Since the latter are linearly independent, neither
J nor AJ occurs in the I, I1 of the leader (59). Also, A and
ala2 occur only in the combinations A + 1, alca2 + 1, since (57)
has no constant term. The coefficients of X3W in L", AIL,
(A + A)LP are respectively
(63) b3+ a102 + 1, Ab3, A + Ab3 +b3aa02,
* AJ is not retained in Ii, since b3AJ - 0, AJ being (34).