2 
MATHEMATICS: L. E. DICKSON 
duced by Hurwitz.^ Although a remarkably simple theory of modular 
invariants has been given,^ no headway was made towards a theory of 
formal invariants before the very recent discovery^ of a simple effective 
method for their construction. This method will be illustrated for the 
linear form / and the modulus 2. The real points (i.e., those with inte- 
gral coordinates) modulo 2 are (1, 0), (0, 1) and (1, 1). The values of 
/ at these points are a, b, a b. Any real linear transformation induces 
a permutation of these three values since it merely permutes the three 
real points. Hence any symmetric function of these three values is a 
formal invariant of /. The elementary symmetric functions reduce 
modulo 2 to zero, i = -\- ab -\- ¥ and j = ab (a -\- b). We pass to 
modular invariants by taking a and 6 to be integers modulo 2. Then 
J = 0, i = / + 1, where / is the invariant in § 1. 
In treating similarly the formal invariants of / modulo 5, we would 
employ the symmetric functions of the fourth powers of our values a, 
b, a -\- b, and not the values themselves. For, (1, 0), (2, 0), (3, 0), (4, 0) 
give the same point and yet lead to the values a, 2a, Sa, oi I; we take 
the fourth power to secure a value uniquely defined by the point. In 
the case of a quadratic form modulo 5, we need only take the squares 
of the values. 
The method is applicable to invariants of several forms in any number 
of variables and to semi-invariants, as shown in the paper cited, which 
gives also a novel method of deriving modular invariants from semi- 
invariants. 
3. Modular plane curves for modulus 2. Let / {x, y, z) be a homogene- 
ous form of degree n with integral coefficients. A point for which the 
three first partial derivatives of / are zero modulo 2 shall be called a 
derived point. If n is even, it need not be a singular point of the curve, 
since it need not lie on the curve ; the argument in the algebraic case, 
based on Euler's theorem 
X — y ~ -\- z — = nf, 
bx by bz 
does not apply modulo 2 when n is even, since the vanishing of the left 
member does not require that of /. A non-singular derived point shall 
be called an apex of the curve; its linear polar is indeterminate. 
For example, any non-degenerate conic (i.e., having no linear factor 
modulo 2) can be transformed linearly into x"^ -\- yz = 0. The only 
derived point is p = (1, 0, 0) and is an apex. Any line through p is 
tangent to the conic; this is evident for z = 0 and follows for y = kz 
since the elimination of y leads to x^ -f- kz^ = 0, with a double root 
