1884.] Non-Euclidian Plane Geometry. 87 



1?* v 



sin 9= =====, 0= = - 



V 1 Zt~ t^ 1 W 



and thence 



X Y=sin0, iX + Y=sin0(0 2 + cosec 2 e), Z=sin0.0. 

 Also 



_0 2 -l + cosec 2 _^ 20 



2 + l + cosec 2 ' 



35 = log cot \Q-\- cos 6, ?/ = sin0cos0, z=sin0sin0. 



10. We have dX?+dY* + dZ z and da; 2 + efy 2 + dz 2 each=cot 2 6>(?0 2 + 

 sin 2 0d0 3 . Writing P, Q = iX Y, iX + Y respectively, we in fact have 

 dX 3 + dY 2 + dZ*= dPdQ + dZ*, where P, Q, Z=sin 0, 3 sin + cosec 0, 

 sin # respectively, and thence 



dZ = sin d d(f> + cos 0cZ0, 



=2 sin 00d0+ (0 2 cos 



giving the formula (ZX 2 + c?Y 2 + cZZ 3 =cot 2 ^^ 2 + sinW0 2 ; and then 

 also dx* + dy 2 + dz*= dx* + (d sin 0) 2 + sm 3 6(Z0 3 = (cos 2 cot 2 + cos 3 6) de~ 

 + sin 2 0(Z0 2 ==cot 3 0d0 2 + sin 2 0d0 3 . Joining to these the differential ex- 

 pression in u, v we have 



- - - - 



(i ^ v z y 



where the final equation d>K. z + dY* + dZ 2 =dx 2 + dy' 2 + dz 2 , shows that 

 the imaginary sphere X 2 + Y 2 + Z 2 = 1 can be bent into the pseudo- 

 sphere. 



Observe that to given values of 0, there corresponds a single point 

 on the pseudosphere, but not conversely, for if 6, be values corre- 

 sponding to a given point, then corresponding to the same point we 

 have 6, + wjr, where n is an arbitrary integer. 



11. The geodesies of the imaginary spherical surface are, of course, 

 its plane sections, any such section being determined by a linear 

 equation aX + ^Y-j-^/Z^O, between the coordinates X, Y, Z. Since 

 for a point corresponding to a real point of the pseudosphere X is a 

 pure imaginary, while Y and Z are real, we see that for a geodesic 

 corresponding to a real geodesic of the pseudosphere we must have 

 x a pure imaginary, /3 and 7 real ; and, in fact, writing as above, 

 P = i'X Y, Q=iX + Y, and therefore conversely X=^'( P Q, 



