to Non-Metrical Vector Algebra. 117 



Such being the case we are free to fix the standard or 

 unit vectors, say a, b, c, d, etc. upon the rays emanating from 

 their common origin quite at our pleasure, choosing them 

 conventionally and independently of one another. There 

 is but one limitation to this freedom, viz. if the ray V is the 

 continuation of the ray /, and if a= Oa has been chosen as 

 the unit vector for /, the unit vector a'= Oa' for V is to be 

 such that, with reference to the fixed 7-line, 



a'= —a. 



This means that, if T a be the terminus of the straight (I' 01) 

 in question, a' ', 0, a, T a should be a harmonic range, with 

 a' , a as conjugates. 



Keeping this only condition in mind let us draw round 

 some closed continuous curve in order to standardize at once 

 the whole pencil of our vectors. Such a curve could be drawn 

 in a variety of ways. It will be remembered, however, that 

 its only office is to enable us to describe numerically the 

 geometrical properties of the plane in as convenient a manner 

 as possible. Now, with this aim in view, it will be found 

 particularly convenient to choose as such standard curve a 

 conic which, as is well-known, can be generated by purely 

 projective processes. 



Let, therefore, 0, the origin of all our vectors, be sur- 

 rounded by some conventionally fixed closed* conic (ellipse) 

 which we will call the standard or the unit conic, and which 

 will be denoted by k. 



Suppose the 2-line has not yet been selected. Then by 

 the very act of surrounding with the conic k and declaring 

 it to be our unit conic (i. e. such that all its radii vectores Oa, 

 Ob, etc. are unit vectors) the T-line is co-determined. In 

 fact, any secant a' Oa drawn through 0, which will hereafter 

 be called a diameter of k, meets the required 7-line in a 

 point T a which is the fourth harmonic to a' ', 0, a, conjugate 

 to 0. In short, the Mine to match the standard conic is 

 the polar of- with respect to k. Thus, having drawn k, 

 draw through (which will be called the centre of k) any 

 two diameters aa' and bb' (fig. 1) ; let ab and b'a' cross in 

 AT, and b'a and a'b in N ; then the join d/i\ 7 will be the 

 required 7 -line, supplementing our reference system. The 

 origin being within the conic, its polar, the 7 -line, will 

 lie outside tc. 



We shall denote all unit vectors, such as Oa or Ob, by 

 small clarendons, a, b, etc., using capitals. A. B, etc. for any 



* /. e. such that no tangents can be drawn from to the conic. 



