145 



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



geodesic circles u = ki 0- ki=rC. A system of concentric circles with the 

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



y2 j^ o^e-'" <■ — 2ev -f e^ — 0^/<x ^ = 

 If X .ji = C we get a system of circles through the origin 



x=+ y2 — 2,^'x = 

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

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



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

 ou the line y = k, O e<k O envelops a unicursal quartic of the form, 

 Ay- f A.x-'^ + A,x=y-' + 2A,x-y + 2A,xy-' + 2A,xy = 



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

 real point and the quartic curve to the geodesic envelop 

 e-2u/c(A-f- A,v2 + 2A,v) -f e-", c(2A,C-'v- + 2A5C-'v) ^iA,/O^)Y' = 



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

 so that tlie quartic envelope (which in this case 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. 



Tlie orthogonal systems cf circles, 



x2 -f y 2 — 2;ix -p b- = 



x^ + (y — h)- = h2 + b-' 



having the radical axis correspond to the geodesies 

 v^ + G=e-=" c — 2,h + b-' = 



and tlieir orthogonal geodesic circles 



V- + O^e--" c _ 2hOe-" 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 Hoiv 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] 



