144 



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

 go over into the circles, 



(X — /3)2 -f- y' = OVa - (See Note 1) 

 and (b) y:=|y_x2)V2 ■> ^g^ 



X = X I 



which changes the circles into the straight lines, 



y = 2;3x — /if^-f CV<x = (10) 



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

 stant go into the parabolas x- = y + constant. The whole 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 



x' + y- — 2/?x 4- k- = 



and their orthogonal trajectories, 



x' + y' — 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 ^ C y = C e 

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

 and X- = y — C- e-. The circles of (8) tangent to the line y = C 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-,:i)'^ + y2 = CVk^ 

 corresponding on the surface to the geodesies, 



\' + C-e-'"'c—2i3y idr- — G'k') —Q -. „ k = e 



Note 1.— Bianchi, p. 419. 



Note 2. — Salmon'-s Conic Sectious. p. 100. 



