178 
MATHEMATICS: F. L. HITCHCOCK 
operational concepts and, if it pleases us, define a vector function as a 
set of three algebraic forms X,Y,Z, homogeneous polynomials in three 
variables x,y,z, with nothing left of the original idea of a vector as a di- 
rected quantity except a definite order in writing the three forms 
X,F,Z. The occurrence of the same mathematical entity under such a 
variety of names and algorisms is perhaps the natural consequence of 
its fundamental character. 
From whatever point of view we prefer to start, it is well known 
in the linear case that a convenient classification of types may be 
made with reference to the axes of the function; an axis of a hnear vec- 
tor function </> of a vector p being a direction of p such that 4>p and p 
are parallel, or <^p = gp, where g is a mere number. In the language 
of algebraic forms this is the same as saying that an axis is a point, in 
homogeneous coordinates, satisfying the equations 
yZ ~ zY = 0, zX - xZ 0, xY - yX = 0. (1) 
In the longer work of which the present paper is an outline, a similar 
basis is taken for a classification of types of quadratic vector functions 
Fp of the vector p. Related mathematical problems which, by reason 
of their close kinship, suggest the study of vectors of higher degree 
are numerous. For example, if x,y,z, and X,F,Z, denote points re-, 
spectively in a first plane and in a transformed plane, the vector Fp 
obviously enough defines a geometric point- transformation. The 
worker who Hmits himself, however, to such an interpretation in homo- 
geneous coordinates will lose sight of the conveniences of vector addi- 
tion. We may with equal ease let Fp define a transformation in space 
of three dimensions with the origin invariant. 
As another application, the properties of Fp, by reason of their 
invariant character with reference to change of coordinate axes, are 
intimately connected with the whole theory of a set of three algebraic 
forms. That the study of the linear vector function led to the dis- 
covery of various invariants belonging to one function or to a system of 
several such functions, is well known. ^ 
Again, the student of certain types of differential equation will find 
that the notion of a vector function comes readily into his work. The 
very appearance of equations like 
dx _ dy _ dz 
where X, F, and Z are algebraic forms as already explained; or like 
{yZ - zY)dx + {zX - xZ)dy + {xY - yX)dz = 0; (3) 
