1884.] Non-Euclidian Plane Geometry. 99 



mining on the unit-circle a point belonging to the angle *c=57 31'; 

 at this point we draw parallel to the tangent a line which is the 

 tangent at the lowest point ; the curve passes through the point 90 

 on the unit-circle, and there cuts the circle at the angle jc=57 31' 

 (or, what is the same thing, the radius at the complementary angle), 

 and we have thus the tangent at the point 90 of the unit-circle ; it 

 will be noticed that this meets the tangent at the apse at a point 

 near to this apse, so that the arc as determined by the two tangents is 

 for a large part of its course nearly a right line ; this is still more the 

 case for smaller values of or K, while for larger values the devia- 

 tion increases, but in the neighbourhood of the unit-circle the form 

 is always nearly rectilinear. 



I show in the same figure the form of the curve for =300, 

 =5-2359877, =tan*, that is c=79 11', r- cos-c=0-1876670, the 

 value at the apse : the construction for the tangent at the unit-circle 

 is the same as before, but in order to lay down the curve with 

 tolerable accuracy we require also the value of r at the node on the 



cos K 



meridian 0=180 ; this is, of course, given by r= /- , that 



v 1 TT^COS^/C 



is r= c08<c , if 7rcosK=sina; whence without difficulty r=0 '23236, the 



cos a, 



value at the node. 



24. The curves shown in the figure are projections upon the plane 

 of the unit-circle, viz., they are the projections on this plane of the 

 geodesies, which cut at right angles a given meridian - T but bearing in 

 mind the form of the meridian, it is easy, by means o the projection, 

 to understand the actual forms on the surface of the pseudosphere. 

 A point near the centre of the figure represents a point high up on 

 the surface ; and in any radius the portions near the centre are the 

 more foreshortened in the figure, and represent greater distances on 

 the surface. Each geodesic cutting the meridian at right angles 

 at the apse descends symmetrically on the two sides, reaches ulti- 

 mately it may be after many convolutions the unit-circle ; the 

 meridian itself is a limiting or special form of geodesic. The unit- 

 circle is not properly a geodesic, but it is an envelope of geodesies. 



25. To obtain all the geodesies we have to consider the geodesies 

 which cut at right angles a given meridian ; and then to imagine this 

 meridian (with the geodesies which belong to it) turned round so as 

 to occupy successively the positions of all the other meridians. The 

 same remark applies of course to the projections ; the figure shows 

 the projections cutting at right angles a given radius of the circle ; 

 and this radius (with the projections belonging to it) is then to be 

 turned round so as to occupy successively the positions of all the 

 other radii. We may imagine the several geodesies turned round 

 separately, each through a different angle, so as to bring each of 



H 2 



