372 



HITCHCOCK. 



INTRODUCTION. 



In this paper it is shown that three ternary quadratic forms X, Y, 

 and Z, homogeneous polynomials of the second degree in the variables 

 X, y, and z, can in general be thrown into the form 



X = VWi — ViW -\- tX, Y = will — WiU -\- ty, Z = ■H2)i — UiO + tz, (I) 



where the seven letters: u, v, iv; Ui, Vi, wi; and t; — denote linear 

 forms, that is, they are homogeneous polynomials of the first degree 

 in X, y, z. 



In the language of vector algebra, the scalars X, Y, Z are the com- 

 ponents of a vector F{p), or simply Fp. The linear forms u, v, and ic 

 are components of a linear vector 4)p, and Ui, V\. Wi are components of 

 a second linear vector dp. The linear form t is regarded as the scalar 

 product of a constant vector 8 and the point-vector p. The above 

 statement translates into the vector equation 



Fp — V4>p 6p -{■ pS8p, 



(11) 



or, in words, a quadratic vector function, homogeneous in p, can be 

 expressed as the vector product of two linear vector functions, aside 

 from a properly chosen scalar multiple of the point-vector. 



The significance of this result lies, in part, in the fact that, in various 

 problems depending on three quantics, the term in p may be taken 

 arbitrarily. For example, the differential equation 



(yZ - zY)dx + (zX - xZ)dy + {xY - yX) dz = 



in vector language becomes ISpFpdp — 0, or SdpVpFp = 0, and is 

 thus independent of the term in p and of the linear form t. The 

 vector VpFp may in general be factored into VpV4>pdp. 



It is also shown that the vector 5, equivalent to the linear form t, 

 may in general be determined in thirty -five ways. The three scalar 

 equations (I), equivalent to the vector equation (II), (A', Y, and Z 

 being given and the right members to be determined), are in general 

 equivalent to eighteen quadratic equations. The solution depends 

 upon an equation of the seventh degree, which determines seven sets 



