io CONCRETE REPRESENTATIONS OF 



It is usual to take k=%i so that the angle between two rays 

 which form one straight line is Jfclog 1=J* . 2imr=mTr. This 

 corresponds to the circular system of angular measurement, 

 and we see that the angle between two rays is periodic, with 

 period 27r. The angle between two lines with undefined 

 sense has, however, the period TT. If the two lines are conjugate 

 with respect to the absolute, (pg, xy)= l, and the angle is 

 JTT. The two lines are therefore at right angles. 



An analogous definition is given for the distance between 

 two points. On the line I joining two points P, Q there are 

 two points belonging to the absolute considered as a locus, 

 viz., the two points of intersection with I. Call these X, Y. 

 The distance (PQ) is then defined to be 



#lo g (P<?, XT) 



where K is a constant depending upon the linear unit em- 

 ployed. 



5. To test the consistency of these two formulae for 

 distance and angle it is sufficient to show that a circle, defined 

 as the locus of a point equidistant from a fixed point, cuts its 

 radii at right angles. 



Let the equation of the absolute, referred to two tangents 

 OA, OB and the chord of contact AB, be xy=z 2 . In a line 

 y=mx through take the point P (x, y, z). Let OP cut the 

 conic in X, Y, and the chord of contact in M. Let X (or Y) 

 and P divide OM in the ratios 1 : k and 1 : p. The coordin- 

 ates of the points are : 0(0, 0, 1), M(\, m, 0), X(l, m, k), 

 P(l, m, p). If (OP) is constant, P describes a circle, and we 

 have the cross-ratio 



(OP, XY)= const. =fjL=^ 9 where k^-k^Jm. 



Hence p= *J 



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Also px=z and py=mz. 



