508] 
IN PARTICULAR THOSE OF A QUADRIC SURFACE. 
163 
The Chief Lines are not in general Geodesics. 
10. That the chief lines are not in general geodesics appears most readily as 
follows: 
To find the condition in order that p = const, may be a geodesic, we write in the 
geodesic equation p' = 0, p" = 0 : the equation thus becomes 
Fq'. GV/ 2 - Gq' (2F, - G 1 ) q 2 = 0; 
that is we have 
FG 2 — 2 F 2 G + GGi = 0 
as the condition in order that p = const, may be a geodesic: the condition that it 
may be a chief curve is G' = 0, which is a different condition. 
We have of course, in like manner, 
2EF 1 - EfF - EE2 = 0 
as the condition in order that q = const, may be a geodesic; and E' — 0 as the 
condition that this may be a chief curve. If p — const., q — const, are each of them 
at once a geodesic and a chief curve, then the four equations must all be satisfied, 
viz. we must have 
FG 2 - 2F a G + GG X = 0, 2EF 1 - E,F— EE2 = 0, 
G' = 0, E' = 0. 
Special Form of the Geodesic Equation. 
11. In the case where the curves p = const., q = const, intersect at right angles 
(and in particular when these are the curves of curvature), we have F = 0; whence 
also F 1 = 0, F 2 = 0; and the geodesic equation assumes the more simple form, 
Ep' (— E. 2 p‘ 2 + 2G 1 pq + Gof 2 ) 
-Gq' ( E 1 p'* + 2E2p'q'-G 1 q'*) 
+ 2 EG (p'q"-p"q') = 0. 
[11“. In the case of a surface of revolution we have 
oc=pcos>q, y=p sin q, z =p; 
E is of the form 1 + P' 2 , P' = d p P, where P is a function of p only, and we have 
ds 2 = (1 + P' 2 ) dp 2 +p 2 dq 2 , 
that is 
E=l+P 2 , E 1 = 2P'P", E 2 = 0, 
F = 0, 
G—p 2 , G 1 — 2p , ^ = 0; 
21—2