1884.] Non-Euclidian Plane Geometry. 89 



t 



which is what the equation 



1 , 2 +A , 



= 0, 



becomes on changing only 2 into 3 + 3 a.^ that is, 2 + 27c 2 7r, where 

 & 2 is an arbitrary integer. We have thus through the two points a 

 singly infinite series of geodesic lines ; in general only one of these 

 is a shortest line, but for points on opposite meridians there are two 

 equal shortest lines. 



13. For the distance between two points (0 1} X ), (0 2 , 2 ) on the 

 pseudosphere, taking (X 15 Y 1? Zj), (X 2 , Y 2 , Z 2 ) for the correspond- 

 ing points on the imaginary sphere, and writing as above P 1? Qj= 

 {Xj Y 1? iXj+Yj ; P 2 , Q 2 =iX 2 Y 2 , iX 2 + Y 2 , we have 



cosh = X^X 2 YjY 2 



=sin e l sin ^ 2 {l(0 1 2 + cosec 2 6> 1 ) +l(0 2 2 + cosec 2 ^ 2 ) 1 2 }, 

 =i sin 0, sin ^ 2 (0 2 -0 i 



Observe here that writing 2 , 02=^ + ^, 01 + CZ0J, and therefore 

 small so that cosh=l + ^ 2 , we obtain 



a 2 = sin 2 



agreeing with the expression for dy? + dy z + dz z . If in the form first 

 obtained we write A 1 =cosec 2 ^ 1 , A 2 =cosec 2 2 , we find 



- 



which is a convenient form. 



In like manner, to find the mutual inclination of the two geo- 

 desies 



A! + BJC0 2 + cosec 2 6) + (^0= 0, 



these correspond to the plane sections A 1 P + B 1 Q + C 1 Z=0, A 2 P 

 + B 2 Q + C 2 Z=0, that is (A 1 + B 1 )iX-h(-A 1 + B 1 )Y+C 1 Z=0, 

 (A 2 + B 2 )i'X 2 -t-( A 2 +B 2 )Y + C 2 Z=0, of the imaginary sphere: and 

 we thence find 



C^-SC 



~ v / C 1 2 -4A 1 B 1 

 15. Suppose that the two geodesies meet in the point , : then 



