NON-EUCLIDEAN GEOMETRY 19 



sentation is to be conform. 1 We shall find that this fixes also 

 the distance function. 



First let us find how a circle is represented. A circle is 

 the locus of points equidistant from a fixed point, or it is the 

 orthogonal trajectory of a system of concurrent straight lines. 

 Now a system of concurrent straight lines will be represented 

 by a linear one-parameter system of circles, i.e. a system of 

 coaxal circles. The orthogonal system is also a system of 

 coaxal circles, and the fixed circle belongs to this system. 

 Hence a circle is represented always by a circle, and its centre is 

 the pair of limiting (or common) points of the coaxal system 

 determined by the circle and the fixed circle. 



The distance function has thus to satisfy the condition that 

 the points upon the circle which represents a circle are to be at 

 a constant distance from the point which represents its centre. 

 To determine this function let us consider motions. A motion 

 is a point-transformation in which circles remain circles ; and 

 further, the fundamental circle must be transformed into 

 itself, and angles must be unchanged. 



15. The equation of any circle may be written 2 

 zz+pz+pz+c=0 



where z=x+iy, p=g+if and z, p are the conjugate complex 

 numbers. Now the most general transformation which pre- 



1 C. E. Stromquist, in a paper ' On the Geometries in which Circles are the Shortest 

 Lines,' New York, Trans. Amer. Math. Soc., 7 (1906), 175-183, has shown that 'the 

 necessary and sufficient condition that a geometry be such that extremals are perpen- 

 dicular to their transversals is that the geometry be obtained by a conformal transforma- 

 tion of some surface upon the plane.' The language and his methods are those of the 

 calculus of variations. The extremals are the curves along which the integral which 

 represents the distance function is a minimum, i.e. the curves which represent shortest 

 lines ; and the transversals are the curves which intercept between them arcs along 

 which the integral under consideration has a constant value. Thus in ordinary geometry, 

 where the extremals are straight lines, the transversals to a one-parameter system of 

 extremals are the involutes of the curve which is the envelope of the system. In 

 particular, when the straight lines pass through a fixed point the transversals are 

 concentric circles. 



" Cf. Liebmann, Nichteuklidische Oeomelrie (Leipzig, 1905), 8, 11. 



