1884.] 



Non-Euclidian Plane Geometry. 



thus the angle at which each branch of the loop cuts the meridian 



0=7T, 



22, To obtain another datum convenient in tracing the curve, I 

 write 0=0 = tan K for the value of at the unit-circle; and intro- 

 ducing for the moment the rectangular coordinates X=rsin0, 



vi ^ ru -i a j ^Y rsin0 r 5 0cos0 , ,, 



Y = l > cos 0, then we easily find = - -- -- ; and thence 



dX r cos + '^0 sin 



for the equation of the tangent at the point on the unit-circle, 



/ T i -i N sin n 0aCOS0 n/ . 

 (y l + cos0 ) = - - 22 rQ __r9(iB si 

 cos 0o +0 sin v 



sin 



For the tangent at the point of intersection with the radius 0=0, 

 or say the apse, we have i/=l COSK; and hence at the intersection 

 of the two tangents 



. 

 z=sm + . 



sn 



. 

 sm0 cos 



_ 1 cos K(COS + sin ) 

 sin 0o 0o cos 



which putting therein = tan K becomes 

 cos KJ1 coa(0 ft K )l 



>. 

 cos K) 



where is given in terms of K by the just mentioned equation 

 = tan K. We have T/=! cos K, x= cos K tan |(0 K) for the locus 

 of the intersection of the two tangents ; this is easily seen to be a 

 curve having a cusp at the unit-circle. 



23. Pig. 3 shows the curves for the values 



We construct and graduate the unit-circle; draw to it a tangent at 

 0, and measuring off from a distance equal to the semi-circum- 

 ference, graduate this in like manner in equal parts to 180 ; then 

 to find the curve belonging, for instance, to =90, we join with 

 the centre of the circle the point 90 of the tangent, thus deter- 



VOL. XXXVII. H 



