QUADRATIC VECTORS. 373 



of values of x, y, and z, called axes of the quadratic vector. When 

 these axes are found we can then find 5 and t. 



The thirty-five ways of finding t are diminished in number by 

 various special relations among the constituents of X, Y, and Z. 

 These restrictions are simply expressible in terms of configurations 

 of the axes. It is shown that at least one value of t can be found, 

 that is, that the eighteen quadratics have a solution, except^in two 

 cases. 



Methods of finding 5 or t are worked out for all possible configura- 

 tions of the axes. This is done by means of normal forms, or model 

 vectors, including various classes of quadratic vectors. 



The use made of the technical methods of vector algebra is slight 

 in the first part of the paper, more extended when dealing with condi- 

 tions of multiplicity among the axes. The results, while they were 

 invariably obtained by vector algebra, can be verified in most in- 

 stances by the reader who has only a slight acquaintance with these 

 methods. 



The classification of vectors under normal or type forms is worked 

 out on the following scheme : — 



Class I. Seven distinct axes. Type form (31). 



Class II. At least one axis is of order two or more. These are 

 subdivided into 



1. Vectors having one or more double axes but no triple axis. 

 The double axes may be one, two, or three, in number. Types Art. 16. 



2. Vectors having at least one triple axis but no quadruple axis. 

 Of these, there may be either one, or two, double axes. We may also 

 have two distinct triple axes. Types Art. 25. 



3. Vectors having an axis of order four or higher. These are of 

 two distinct kinds, according as the multiple axis is — 



Class 1°. A double element of all cones VpFp = 0. Type forms 

 are discussed in Art. 27 and 37. 



Class 2°. Not a double element for all cones VpFp = 0. Types 

 Art. 30, including the two forms (1S5) and (195). 



Class III. The number of axes is infinite. There are three special 

 forms (57). The vector VpFp possesses a scalar factor, and is said 

 to be reducible. The equations (II) are not considered with reference 

 to this class, the forms (57) being equally simple. 



