178 
ON GEODESIC LINES, &C. 
[508 
Initial direction M7: the geodesic touches at opposite infinities the mean hyper 
bola - + — = 1, it lies wholly in front of the plane y = 0 of this hyperbola. 
ct c 
Initial direction M8: the geodesic touches a hyperbolic curve of curvature parameter 
q 8 where q 8 (negative) is between — b and q 9 the parameter of the hyperbolic curve 
of curvature through M; viz. it cuts all the hyperbolic curves the parameters of which 
are between — b and q 8 , but does not cut the remaining curves the parameters of 
which extend from q 8 to — a. 
Lastly, initial direction is M9, that of the hyperbolic curve of curvature through M; 
the geodesic touches this curve, cutting all the hyperbolic curves the parameters of 
which are between — b and q 9 , but not any of those the parameters of which are 
between q 9 and —a. 
37. If in the differential equation of the geodesic line we consider p, q as the 
elliptic coordinates of a given point M of the curve, the equation for a given value 
of 9 determines the direction of the curve; or conversely, if the direction be given, 
the equation determines the value of the parameter 0. Writing 
V(a + p)(b +p) (c + p) ’ V(a + q) (b + q)(c + q) 
then P, Q are proportional to the rectangular coordinates of a consecutive point M', 
measured from M in the directions of the hyperbolic and oval curves of curvature 
respectively; and the differential equation of the geodesic lines gives 
Vp + 9 \!q + U 
viz. if </> be the inclination of the geodesic to the hyperbolic curve of curvature, then 
<f) = 0, 0 = — q, = oo , 0 = —p, as it should be.