374 HITCHCOCK. 



PART ONE. 



1. In many problems of Geometry and of Physics we meet with 

 quantities which, occurring at the points of a portion of space, possess 

 a definite direction, as well as magnitude or length along that direc- 

 tion. The velocity of a fluid at a point, and the force at a point due 

 to the attraction of an assigned distribution of matter, are familiar 

 examples. If such a directed quantity, or geometrical vector, be 

 resolved along chosen axes of reference, it yields components X, Y, 

 and Z, which are ordinary, or scalar, functions of x, y, and z, the 

 coordinates of a point in space. Approaching the matter from the 

 side of Algebra, both the independent variables and the component 

 functions will, in general, be free to take on complex values. 



If we agree upon the following four conventions, — 



1. The vectors whose components are (1, 0, 0,), (0, 1, 0), and 

 (0, 0, 1) are denoted, respectively, by i, j, and k. 



2. Multiplication of a vector by a scalar means multiplication of 

 all components of the vector by the scalar. 



3. Vectors are added by adding their corresponding components. 



4. Equality of vectors implies equality of corresponding compo- 

 nents, — 



it follows that we may write, for the vector p from the origin to a point 

 in space, 



p = ix + jy + kz (1)- 



A vector function of p may be denoted by F{p), whence 



F{p) = iX + jY + kZ (2). 



My present object is to contribute something toward a theory of 

 those vectors whose components are homogeneous polynomials 

 of the second degree in the variables x, y, z. That is, X, Y, and Z 

 are ternary quadratic forms. The theory of my vectors will then 

 be, up to a certain point, in close relation with the theory of a set of 

 three forms, but the name vector implies also a definite order, X, Y, Z, 

 among these forms; and we have, moreover, the above defined 

 process of addition of vectors. 



