147 



The equations of transformation from the plane xy to the plane 5 T — plane 

 are, 



x = ( o (1 -: ) 



y = (( : (!-:))-( r-} (1-: )')'/' 



and the equations of transformation from the pseudosphere to this plane 



are, 



v= + c^e-^" c = ( : ) (1 — : ) 



Discussion op the Transformation. 

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



p -\- r- - : = 



The circles x - -j- y " — 2/3s + /^ ^ — C ^ ac - ^= become the straight lines 

 (a = — -i-x - + = ) : — 2 -)'x - ^^- ,>2x 2 _ (32 -_ 

 The straight lines y ^ k go into a sheaf of conies, 



(k2 -J- l)r- — (2k2 + 1); -^ ^^2 + k^ irz through the point 

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

 ses. The real part of the pseudosphere 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 £ — 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) (a^ +b- + 1) 

 := (a- + b2)/(a- + b^ + 1) 

 (2) 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 and 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. 



