. 145 



These may be regarded as a system of geodesies having as an envelop the 



geodesic circles u = ki 0<ki=C. A system of concentric circles with the 

 centers at any point (e, 0) on ox gives the geodesies 



v2 -f C2e-2" c _ 2ev -f e^' — C/oc ^ = 

 If X ./3 = O we get a system of circles through the origin 



x2-|- y2 — 2;3x = 

 which correspond to a system of geodesies through a point. In this ease, 

 however, the point is not a real point of S. 



A system of circles with the centers on ox and passing through a point 

 on the line y — k, O e<k<C envelops a unicursal quartic of the form, 

 Ay- ^ A,x2 + A,x==y2 + 2A 3X =y -f 2A ,xy ^ + 2A ,xy =r 

 This system of circles corresponds to a system of geodesies through a 

 real point and the quartic curve to the geodesic envelop 

 e-2„/c(A-<- A,v- + 2A,v) -1- e-" c(2A30-iv2 + 2A5O-1V) -p (Ai/O2)v2=:0 

 In this case the circles have a second common point on the line y = — k 

 so that the quartic envelope (which in this ease is imaginary), having four 

 nodes, breaks up into two circles which are themselves curves of the sys- 

 tem and therefore correspond to the geodesies of the surface. 

 The orthogonal systems cf circles, 

 x2 + y2 — 2;3x-r b^ =zO 



x= + (y — h)= = h2 -4- b^ 



having the radical axis correspond to the geodesies 



v= + C=e-^-" c — 2Jv + b- = 



and their orthogonal geodesic circles 



V' + O^e--" c — 2hCe-" c + b^ = 



These may be such that the limiting points of the circles are real and 

 distinct, coincident or imaginary. It is interesting to note that this sys- 

 tem of circles, which in so many problems in applied mathematics repre- 

 sents lines of flow and equipotential lines may be mapped conformally on 

 the pseudosphere in such a way that the lines of flow and the equipoten- 

 tial lines are the geodesies of a system and their orthogonal geodesic cir- 

 cles. 



Another straight line representation of the geodesic lines of the sur- 

 face S. 



[10—18192] 



