147 



The equations of transformation from the plane xy to the plane i C — plane 

 are, 



x= ( f ) (1 -: ) 



y = (( : (1 -:))- ( ^^)(i-c )2)V'2 



and the equations of transformation from the pseudosphere to tliis plane 

 are, 



V= ( f ) (1- C ) 



Discussion of the Transformation. 

 The entire upper part of the xy— plane is represented inside the circle 



f2 + r- -- : = 



The circles x ^ -f y 2 _ 2,?x 4/32 _ o ^ ac = = become the straight lines 



(oc 2 — r-oL ^ + c^) : — 2 ;S'3c 2 fa- ,923c 2 __ 02 = 



The straigh t lines y = k go into a sheaf of conies, 



(k2 -f- 1):2 _- (2k2 + 1): + ^2 ^ ].2 _ through the point 

 (0, 1). And since — (k2 -\- 1) is always negative the conies are all ellip- 

 ses. The real part of the pseudospliere is therefore represented in the area 

 included between the ellipses corresponding to the lines y = C and y = C/e. 



All the ellipses are tangent to the cir- 

 cle at the point (0, 1) and have their foci 

 on the C — axis. The circles concentric at 

 the origin become the lines C = constant, 

 chords parallel to the f — axis. The system 

 of circles with centers on o — x and pass- 

 ing through the point a,b goes over into 

 the system of straight lines through the 

 point 



,^ = (a) (a2 + b2 + l) 

 i = (a2 + b2) (a2 + b2 + 1) 

 Q ' Two such systems properly related and 



Fig. 2. having the point (a,b) on the same line 



y ::= b go over into the two projectively related sheaves of lines whose cor- 

 responding rays intersect on the conic corresponding to y =: b. In par- 

 ticular, in case the points (a,b) are on the x — axis the conic becomes the 

 circle o — b aud the corresponding rays are at right angles. Circles with 

 the centers on the x — axis and of equal radii go over into the straight lines 

 enveloping an ellipse. The line x = goes into f = the points being 

 moved along the line. The origin is the fixed point of transformation. 



