144 



which transforms S conformally on the plane so tliat the geodesies lines 



go over into the circles, 



(X — /3|2 ^ y2 = C2 cc = (See Note 1) 



and (b) j=(j— x2)V2 ■ ) ^g^ 



X =1 X I 



which changes the circles into the straight lines, 



J = 23x - (i^-i- C2/(x 2 (10) 



By (9) the x axis goes into the parabola x^ = y and all the lines y = con- 

 stant go into the parabolas x^ ^= y -|- constant. The '.vhole upper part of 

 the plane is represented inside the parabola x^ = y. The points on the 

 lines X = constant are moved along the lines. The origin is the fixed point 

 of transformation. 



Circles concentric at the origin correspond to lines y "= constant while 

 every system of concentric circles on the x axis goes over into a system of 

 parallel lines. A system of circles given by (8) passing through a point 

 corresponds to a system of lines tlirough a point. A system of circles with 

 the y axis as radical axis 



x2-f y2 _2,3x + k2 = 



and their orthogonal trajectories, 



x= + y2— 2hy = + d2 (See Note 2) 



corresponds to a sheaf of lines and a sheaf of conies. 



The geodesies v = constant correspond to the lines x = constant i. e. 

 to the diameters of the parabola x^ =r y. The entire real part of the sur- 

 face S is represented in the xy — plane by the strip y = y =i C/e 

 and in the xy — plane by the strip included by the curves x^ = y — C 

 and X- ^ y — O- e^ The circles of (8) tangent to the line y = O e go over 

 into a system of straight lines enveloping the parabola x- = y — C^.e^ 



Since the representation given by (8) is conformal it is interesting to 

 note that the lines y == constant may be considered as the envelop of a 

 system of circles of constant radii and centers on the x axis given by the 

 equation, 



(x-/3)2 + y2 = CVk = 

 corresponding on the surface to the geodesies, 



v= + C2e-2"'c_23v +(;3= — C^ k^) =0 0<k = e 



Note 1.— Bianchi, p. 419. 



Note 2.— Salmon's Conic Sections, p. 100. 



