156 
[508 
508. 
ON GEODESIC LINES, IN PARTICULAR THOSE OF A QUADRIC 
SURFACE. 
[From the Proceedings of the London Mathematical Society, vol. IV. (1871—1873), 
pp. 191—211. Read December 12, 1872.] 
The present Memoir contains an investigation of the differential equation (of the 
second order) of the geodesic lines on a surface, the coordinates of a point on the 
surface being regarded as given functions of two parameters p, q, and researches in 
connection therewith; a deduction of Jacobi’s differential equation of the first order 
in the case of a quadric surface, the parameters p, q being those which determine 
the two sets of curves of curvature; formulae where the parameters are those which 
determine the two right lines through the surface; and a discussion of the forms of 
the geodesic lines in the two cases of an ellipsoid and a skew hyperboloid respectively. 
Preliminary Formulae. 
1. I call to mind the fundamental formulae in the Memoir by Gauss, “Disquisitiones 
generales circa superficies curvas,” Comm. Gott. recent, t. vi., 1827, (reprinted as an 
Appendix in Liouville’s edition of Monge,) together with some that I have added to 
them. The coordinates x, y, z of a point on a surface are regarded as given functions 
of two parameters p, q, these expressions of x, y, z in effect determining the equation 
of the surface, and we have 
dx + \d?x = a dp + a'dq + \ (a dp 2 + 2a.' dp dq + a" dq 2 ), 
dy + \d 2 y = b dp + b'dq +|(/9 dp 2 + 2 /3'dp dq + /3"dq 2 ), 
dz + \d 2 z = c dp + c'dq + § (7 dp 2 -f 2y'dp dq + <y"dq 2 ), 
A, B, C —be' — b'c, ca' — c'a, ab' — a'b;