110 
MATHEMATICS: 0. E. GLENN 
These concomitants, and their polars hy co = Xi^ h — , form a complete 
8x1 8x2 
scale for Kj,. 
A fundamental system of formal covariants of /s modulo 2 consists of the 
following twenty quantics;^^ where we abbreviate by H the algebraic hessian 
of fs, and 
1 = H-{-{f,){fs},l = QlH]-{-fs{fs), 
and in which L, Q are the universal covariants of the group G^: 
[Ef,.t], {jTi}, {m],[Qn,L,Q, 
if,), m, {th), {tEjz), I- 
The invariant /, which is not' represented as belonging to any scale, is 
I = a(? -\- a^az + a-i^ + (ao + az) (fs). 
The orders of the forms in this system range from 0 to 3, and their degrees 
from 0 to 4. 
The proofs of the existence of {Ku} and of [Kj;] involve certain remark- 
able congruential properties of the binomial coefficients.® 
Reduction methods rest largely upon the fact that two modular covariants 
of the same order and led by the same seminvariant, while not identical as 
a rule, are necessarily congruent to each other moduh L(= XxX^ — XxX^), 
and p. Thus a covariant is not uniquely determined by its seminvariant 
leading coefficient as is the case with algebraic covariants. Another note- 
worthy general fact is that of the existence of covariants whose leading 
coefficients are modular invariants. 
^Hurwitz, A., Archiv. Math. Phys., Leipzig, (Ser. 3), 5, 1903, (17). 
2 Dickson, L. E., Trans. Amer. Math. Soc, 10, 1909, (123); 12, 1911, (75); 14, 1913, (290); 
and 15, 1914, (497); Amer. J. Math., Baltimore, 31, 1909; etc. 
3 Dickson, Madison Colloquium Lectures, 1913. 
* Miss Sanderson, Trans. Amer. Math. Soc, 14, 1913. 
6 Glenn, O. E., (a) Amer. J. Math., 37, 1915, (73); (b) Bull. Amer. Math. Soc, New York, 
21, 1915, (167); (c) Trans. Amer. Math. Soc, 17, 1916, (545); (d) 19, 1918, (109); (e) Ann. 
Math., Princeton, 19, 1918, (201). 
® A paper containing a portion of the general theory here mentioned and the derivation 
of the system of the quadratic modulo 3, as well as certain other developments, is to appear 
in Trans. Amer. Math. Soc. 
