766] 
323 
766. 
ON THE GEODESIC CURVATURE OF A CURVE ON A 
SURFACE. 
[From the Proceedinqs of the London Mathematical Society, vol. xn. (1881), pp. 110—117. 
Read April 14, 1881.] 
There is contained in Liouville’s Note II. to his edition of Monge’s Application 
de l’Analyse à la Géométrie (Paris, 1850), see pp. 574 and 575, the following 
formula, 
1 di 1 dG 1 dE . . 
p~ ds + 2G^/E du C0SÏ 2Ef/G dv Smh 
di cos i sin i 
= — —I—-—— H , 
ds p 2 p 1 
which gives the radius of geodesic curvature of a curve upon a surface when the 
position of a point on the surface is defined by the parameters u, v, belonging to 
a system of orthotomic curves ; or, what is the same thing, such that 
ds 2 = Edu- 2 + Gdv 2 . 
Writing with Gauss p, q instead of ti, v, I propose to obtain the corresponding formula 
in the general case where the parameters p, q are such that 
ds 2 = Edp 2 + 2 Fdpdq + Gdq 2 . 
I call to mind that, if PQ, PQ' are equal infinitesimal arcs on the given curve 
and on its tangent geodesic, then the radius of geodesic curvature p is, by definition, 
a length p such that 2p. QQ' = PQ 2 . More generally, if the curves on the surface 
are any two curves which touch each other, then p as thus determined is the radius 
of relative curvature of the two curves. 
41—2