158 
ON GEODESIC LINES 
[508 
the differential equation of these is (E', F', G'\dp, dq) 2 = 0; and in particular if 
E' = 0, G' = 0, then this becomes dp dq = 0, or we have p = const., q = const, for the 
equations of the two sets of chief curves respectively. On the hyperboloid the chief 
curves are the two sets of generating lines. The chief curves are not in general 
geodesic lines, but on the hyperboloid, qua straight lines, they are, it is clear, geodesic 
lines. 
3. The directions of the curves of curvature, or the principal tangents, and the 
corresponding values of the radius of curvature are determined by 
p _ Edp + Fdq _ Fdp + Gdq 
V ~ E'dp + F'dq ~ F'dp + Gdq 5 
or, what is the same thing, these directions are determined by the equation 
dq 2 , 
— dq dp, 
dp 2 
E , 
F , 
G 
E', 
F' , 
G' 
The same equations may be written 
, , pE' - VE P F' - VF 
dq : dp pF ,_ y F - pG ,_ y G ; 
that is the principal radii of curvature are determined by the equation 
p 2 (E'G' - F' 2 ) - P V(EG' + E'G - 2FF') + V 2 (EG - F 2 ) = 0, 
(last term is = V 4 , but it is better to retain the original form): and then, p being 
either root, the last preceding equations give the direction of the curve of curvature 
corresponding to the given value of the radius of curvature. 
If p — const., q = const, are the equations of the two systems of curves of curvature 
respectively, then the quadric equation in (dp, dq) must become dpdq= 0; this will be 
so if F=0, F'= 0; and we thus have these equations, viz. written at full length 
they are 
d p x d,f + d p y d q y + d p z d q z = 0, 
dpX , d p y , dpZ 
dqX , d q y , d q z 
dpdps, d p d q y, d p d q z 
= 0, 
as the conditions in order that p = const., q = const, may be the two systems of curves 
of curvature. The former of these equations merely expresses that the two sets of 
curves always intersect at right angles.