American Journal of Mathematics, Vol. 51, No. 1 (Jan. 1929), 123-138

Presented to the American Mathematical Society, November 27, 1926.

Introduction

This paper considers a generalization of the symbolic method of Masehke for representing the invariants of a quadratic differential form Σg_{ij}dx^{i}dx^{j}. According to this method, the coefficients g_{ij} are represented symbolically as products of f_{i}f_{j}.

Maschke regarded f_{i} and f_{j} as partial derivatives, ∂f/∂x^{i}, ∂f/∂x^{j}, of a symbolic function f, and placed

(1) A = Σg_{ij}dx^{i}dx^{j} = (df)^{2} =
(Σf_{i}dx^{i})^{2}.

The possibility of a similar symbolic representation in which it is not assumed that ∂f_{i}/∂x^{j} = ∂f_{j}/∂x^{i} has been suggested by Wilson and Moore. This is the generalization that is to be considered in this paper.

As might be expected this generalization does not affect the mode of representing first order differential parameters. It is found, however, that invariant forms of higher order can best be regarded as invariants of the set of quantities g_{ij} together with certain other sets.

The two dimensional case is discussed in some detail. A geometric interpretation is introduced, and certain invariant vectors related to a surface are studied.

The f_{1}(u, v) and f_{2}(u, v) are here regarded as any pair of independent vectors, satisfying the relation (1), associated with the point u, v of the surface. We find that the plane (called the base plane) which they determine has many properties analogous to properties of the tangent plane in the ordinary case. The effect of varying the orientation of the vectors f_{1} and f_{2} in the base plane is considered only in a special case.

Normals to this plane which are expressible in terms of the first derivatives of f_{1} and f_{2} are called the first normals relative to the base plane. In general, there are four independent first normals to the base plane instead of three as in the usual case.

The analogues of the various curvature vectors associated with the curves a(u, v) = constant are studied. These vectors in the main have properties very similar to the properties of the usual curvature vectors. There are, however, certain novel features. In particular we mention the appearance of a new invariant vector.