to Non-Metrical Vector Algebra. 119 



will again be a conic, ./e n homologous with the standard conic. 

 For 0<r<l, all these conies will be closed and contained 

 within k. For r>l, however, these conies will not continue 

 for ever to be closed. 



When r reaches a certain value, say r , k will be a parabola, and for 

 r>f^ the conic will either at once split into two branches, a new branch 

 appearing "beyond" the T-line, or else we shall continue to have one- 

 branched curves up to a certain value r h > r , and only when r h is ex- 

 ceeded, two-branched conies (hyperbolas). The former will be the case 

 for the euclidean, and the latter for a lobatchevskyan space. (This, 

 behaviour is closely connected with the peculiarities of the construction 

 of the negative of a vector, cf. P.V.A. 9.) But details of this kind 

 being irrelevant for our purposes, need not detain us here. For r- >cc 

 the conic k } . will tend to a pair of straight lines falling, so to speak, into- 

 the T-line from both sides. (If space is elliptic, the T-line, as every 

 straight, has but one " side " ; in such cases, as in P.V.A. , let us con- 

 template a "restricted region " broad enough to embrace all the contem- 

 plated figures.) 



Let <r, y be the (non-homogeneous) projective coordinates 

 of a point of k with respect to as origin and a, b as axes,, 

 so that 



r = ^a + ?/b, 



which reads : the end-point of r is reached from by ar 

 projective unit steps along the a-line, followed by y steps 

 along b, or vice versa (addition being commutative). Then, 

 remembering that | a | is to be the unit of a?, and | b | that of y, 

 the equation of our standard conic with inspect to any axes 

 a, b whatever, can be written 



x 2 + y 2 + i lc l2 xy = l. 



The value of the coefficient c 12 will depend upon the choice 

 of the particular pair a, b as axes. It will be characteristic 

 for the pair of rays represented by a, b or, if we prefer to 

 put it so, for the " angle a, b." To bring this into evidence 

 let us better introduce, instead of c 12 , the symbol 



(ab) or (ba), 



indiscriminately, since the roles of a and of b in making up 

 the scalar number c 12 are manifestly the same. Both (ab) 

 and (ba) are hitherto mere synonyms of c 12 , the coefficient 

 of 2xy, and nothing else. After this warning we can write* 

 for the standard conic, 



# 2 + 7/ 2 + 2(ab>// = l, (1) 



with respect to any (distinct) a, b as axes. How to unravel 



