1884.] Non-Euclidian Plane Geometry. 101 



QB, QC meets the line BC ; while any line through Q extramediate 

 to QB, QC does not meet the line BC. 



27. It is interesting to connect the theory of the geodesies of the 

 pseudosphere with the general theory of geodesies. Starting with 

 the form 



we have E=cot 2 0, F=0, G=sin 2 0; and therefore E-f 1 = -L or 



G 



E= - 1, and the differential equation of the geodesic becomes 



E0' . 2G^'0' - G^E^-G^' 2 ) + 2EG(0'0"-0''0')=0 ; 

 that is 



T-, dE n dG 

 where E 1= _, G 1= _, 



and writing here E = 1 , 



we have E^-' 2EG 1 -E 1 G=G/ -2\ 



G 3 \G / 



Moreover, from G^sin 2 ^ we find G 1 =2v G.I G, and the equation 

 becomes 



Introducing here G in place of by the equation G=sin 2 0, we have 



"2v/G.l-G' 



"_ 1 

 4G*(1 G 



and the equation thus becomes 



-G)G" + (l-2G)G /2 }0'=0. 



The whole term in 0' is thus 0'{-2G(l-G)G"+(4-4G)G' 2 }, 

 which divides by 2(1 G); the whole equation thus divides by 2(1 G), 

 and omitting this factor, the equation becomes 



0'(2G /2 -GG") +20'3G 8 +0"GG'=0, 



