164 
ON GEODESIC LINES, 
[508 
hence the differential equation is 
(1 + P' 2 ) [p 2 (p'q' -p'q) + -pY*q'P'P" +p 3 q' 3 = 0 ; 
this has an integral 
(1 + P' 2 )p' 2 11 
p4q'2 p2 (J2 > 
or say 
G 2 s' 2 = p 4 q' 2 
where 
s 2 = (1 4- P' 2 )p' 2 +p 2 q' 2 . 
Writing here p, yjr for p, q, where p is the distance of the point from the axis, 
and yjr is the longitude reckoned from an arbitrary meridian, then the equation is 
Gds = p-dÿ, 
which is the equation given by Legendre, Théorie des fonctions elliptiques, t. I. p. 361. 
This may also be written 
— = cos 7 
P 
if 7 be the inclination of the geodesic line to the parallel of latitude.] 
Geodesics on a Quadric Surface. 
rjQ-2 /£/2 ^2 
12. In the case of a quadric surface - + ~ + -- = 1, writing for shortness 
a, ¡3, 7 = h — c, c — a, a — b, we may express the coordinates x, y, z in terms of two 
parameters p, q as follows: 
— /3y P = a (a +p) {a + q), 
-yay 2 = b (b +p) (b +q), 
— afi z 2 = c (c + p) (c + q), 
where, in fact, p = const., q = const, are the equations of the two sets of curves of 
curvature respectively. Writing moreover 
p _ E Q — 1 
(a +p) (b +p) (c +p) ’ (a + q) (b + q) (C+q) ’ 
we have 
ds 2 = j(p- q) (Pdp 2 - Qdq 2 ), 
that is 
E, F, G=\(p-q)P, 0, l(q-p)Q; 
and the geodesic equation becomes 
Pp' {.Pp' 2 - 2Qp'q' + (Q + q ~p Q') q /2 } 
+ Qq' {(P+p-q P')p' 2 - 2Pp'q' + Qq' 2 } 
- 2 (p~q)PQ (p'q" - p 'q ) = 0,