157 
508] 
ON GEODESIC LINES, &C. 
whence differential equation of surface is 
Adx + Bdy + Cdz = 0. 
Also 
E, F, G = a- + b 2 + c 2 , o,of -f bb' + cc / , cl 2 + b ~ 4* c5 
so that element of length on the surface is given by 
dor} 4- dy 2 4- dz 2 = Edp 2 + 2 Fdp dq 4- Gdcf; 
= (E, F, GQdp, dq) 2 ; 
V 2 =A 2 + B 2 + C 2 = EG-F>. 
or, as I write it, 
and moreover 
The equation (E, F, G][dp, dq) 2 = 0 determines at each point on the surface two 
directions (necessarily imaginary) which are called the “circular” directions. Passing on 
the surface from point to point along the circular directions, we obtain two series of curves 
(always imaginary) which are the “ circular ” curves; the equation (E, F, G^dp, dq) 2 = 0 
is the differential equation of these curves; and if we have E = 0, G = 0, then this 
becomes dp dq = 0; viz. we have in this case p = const, and q — const, as the equations 
of the two sets of circular curves respectively. It is clear a priori, and will be shown 
analytically in the sequel, that the circular curves are geodesic lines. 
I write also 
E', F', G' = Aa + B/3 + Gy, Aa' + B(3' + Gy', Aa" + B/3" + Gy", 
or, what is the same thing, 
E', 
F', 
G' 
represent 
the 
a, b, 
c 
? 
a, 
b , 
c 
y 
a', b', 
c' 
a', 
V, 
d 
ct, ¡3, 
7 
/ 
« 
¡3', 
! 
7 
a , b , c , respectively, 
a , b' , c' 1 
a , P , y 1 
(these last symbols do not occur in Gauss). [They are the D, D', D" of Gauss.] 
2. The radius of curvature of normal section corresponding to direction dp : dq is 
given by 
p (E, F, G\dp, dq) 2 
V~(E', F', GJdp, dq) 2 ’ 
whence it appears that the directions of the inflexional or chief tangents (the Haupt- 
tangenten) are determined by the equation 
(E, F\ G'\dp, dq) 2 = 0. 
The directions in question are imaginary on a surface such as the ellipsoid where 
the curvatures are in the same direction, but on a concavo-convex surface they are 
real; and in particular on the hyperboloid they coincide with the directions of the 
generating lines. We may on any surface pass from point to point along the chief 
directions; we have thus on the surface two sets of curves which are the chief curves j