MATHEMATICS: O. E. GLENN 
109 
A scale of modular concomitants is said to be complete when it contains all 
invariants and covariants of the first degree in the coefficients of K which can 
be obtained from K by empirical or modular invariantive processes, i.e. a 
fx-adic scale is complete when it constitutes a fundamental system of first 
degree concomitants of K. Thus we find that we are able to construct a com- 
plete modular system of a form fj^ by a process of passing from a complete 
scale for of the first degree to scales derived from covariants of of 
degree > 1. 
3. A complete system of the quadratic, modulo 3. — I have proved that the 
following eighteen quantics constitute a complete formal system modulo 3 of 
/2; the proof resting primarily upon the fact that, when p = 3, every co vari- 
ant is of even order, and hence the ix-adic scale (5) is complete: 
h, Ci, im, [/2C1], [f,'C,l Ef,, (6) 
L, Q, im, ([/22p), (Ci£/2), r; 
where^ V = (ao + a^ {2ao + 2ai + a^) {2ao ai-\- c^), and L, Q are the func- 
tions to which the universal covariants of the group G(^2_^)(^2_i) reduce 
when p = 3. These covariants are^ 
L = Xi'x2 — XiX2^, Q = {Xi'^0C2 — XiX2^V L. 
The last four quantics in the system (6) are pure invariants and constitute a 
complete system of invariants of /2 modulo 3. This set of invariants was first 
derived by Dickson. The orders of the forms (6) range from 0 to 6 and the 
degrees from 0 to 6. 
4. A complete system of the cubic, modulo 2. — When p = 2 the m concomi- 
tants (5) do not form a complete scale. Every form of order > 3 is reducible ^® 
modulo 2 in terms of invariants of the first degree in its coefficients and of its 
covariants of orders 1, 2 and 3. UK;, is the co variant shown in §1 we have 
iP=2), 
(K,) = Ci + . . + C,_i, 
[K,] = [Co + {K,)]x, + [{K,) + C,W 
These, and the additional covariant 
{K^} = CqXi^ -f {Kj,)xiX2 + 0^X2'^, 
exist for all orders v. When v is odd there exists also a cubic covariant^'^ 
{Kv} = CoXi^ + I1X1H2 + 12:^1:^2^ + C^X2^, 
where Ii, 1 2 are definite linear expressions in Ci, . . . , C„_i such that^<^ 
/1 + /2 ^ (/v,)(mod 2). 
