1884.] 



Non- Euclidian Plane Geometry. 



95 



we haveaj=log tan (|n--j-^) sin, y=cos <z, where log tan (JTT + ^Z), 

 the hyperbolic logarithm (which has been the signification of log 

 throughout), is the function tabulated Tab. IV. Leg. " Traite des 

 Fonctions Elliptiques," t. ii, pp. 256 259. 



We may hence obtain the values of the coordinates as follows : 



Attending only to one-half of the surface we may regard the surface 

 as standing on the circular base 7/ a + z 2 =l : say this circle is the 

 equator, or the unit-circle : the horizontal section being always a 

 circle, the radius diminishing at first rapidly and then more and more 

 slowly from 1 to as the height increases from to oo . It is 

 hardly necessary to remark that the radius of the equator is any 

 given length whatever, taken as unity : the equations might, of 

 course, have been written = c{log cot ^0 cos 6 } , \/ 7 ,2 _|_~^2 c S { n t 

 but there would have been no gain of generality in this. 



20. The geodesies are as already seen given by an equation 

 A + B(0 2 +cosecV)+C0=0. If B=0, we have A -f 00=0, that is 

 0=const., which belongs to the meridians ; if B be not=0, we may by 

 a mere change of 0, that is, of the initial meridian, reduce the form to 

 + cosec 2 #) = 0, which is the equation of a geodesic cutting at 



right angles the meridian 0=0 ; writing herein sin 0=-, we have 



r 



A + Bl 2 + 1=0, which is the equation in the polar coordinates 



r, of the projection of the geodesic on the equatorial plane x=0 : 

 putting herein for greater convenience B= A& 2 , we have r 2 =: 



j : we require only such portions of the curves as lie within 



~ 



- 



J. ~~~ rC~ 



the unit-circle, and need therefore attend only to those for which 7c is 



