OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
367 
The Independent Magnitudes connected with the Curve. 
32. Various magnitudes connected with the curve (j) = 0 are required; we take 
i = its circular curvature, 
P 
i = its curvature of torsion, 
T 
= the circular curvature 'j 
^ I of the geodesic tangent, 
= the curvature of torsion J 
T 
= its geodesic curvature, 
P 
It =z the radius of the osculating sphere, 
CT the angle ])etween the normal to the surface and the principal normal 
of (f) = 0, and 
B = , 
dn 
where dn is the normal distance at tlie point of ^ = 0 from the curve (f> + d(f) = 0. 
Further, we write 
II 
^ + 2Mx'y' + Nt/'q 
Fx + Fy, 
Fx' + Gt/' 
V 
1 Lx' + My, 
Mx' + N.y' ’ 
N = ^ 
Ex' + Ft/', 
77ix''^ + 2in'x'y' + rn'y'^ + Ex" 
+ F//" 
V 
Fx' -j- Gy, 
nx'-^ + 2/fix'//' + + F./' 
+ G//" 
with the customary notation for rn, rn', rn", n, n\ n" ; then A = 0 gives tlie asymptotic 
lines, W = 0 gives tlie lines of curvature, N = 0 gives geodesic lines. Moreover, 
W3 = AH - A" - K, 
where H and K are the mean curvature and the specific curvature of the surface at 
the point, viz., 
H = ^ ^ , K = ^ , 
Pi P-2 PiP-2 
Pi and being the principal radii of curvature. We have the relations'^ 
* See Stahl und Kommerell, ‘Die Gnmdformeln der allgemeinen Fliichentheorie,’ § 14, for some 
of them. 
