508] 
IN PARTICULAR THOSE OF A QUADRIC SURFACE. 
159 
General Theory of the Geodesic Lines on a Surface. 
4. I now proceed to investigate the theory of geodesic lines on a surface, the 
surface being determined as above by means of given expressions of the coordinates 
x, y, z in terms of the parameters p, q. 
The differential equation obtained by Gauss for the geodesic lines is in a form 
not symmetrical in regard to the two variables ; viz. his equation is 
- dp 2 + 2 dF dp dq + ~ dp* = 2ds d Edp 1~ Fdq , 
dq dp 1 2 dp ds 
where, as above, 
ds 2 = (E, F, Gfdp, dq) 2 . 
If we herein consider p, q as functions of a parameter 0, and write for shortness 
d e p, d e q, dd 2 p, &c. = p', q, p", &c., 
n = (E, F, GJp', q y, 
d p E=E 1 , d q E = E 2 , &c., 
also 
and 
then the equation is 
(E u F„ <?,$*/> q’Y- = 0. 
We have 
(Ep + FqW _ 1 
V fil ) nfn 
(M + N), 
where N is the part containing p", q", which I will first calculate; viz. we have 
N = il W + Fq") - (.Ep + Fq) ±Cl', 
= n (Ep" + Fq") - (Ep + Fq') {(Ep' + Fq’) p" + (Fp' + Gq') q"}, 
= p" {ELL - (Ep' + Fq') 2 } + q"\Fn - (Ep + Fq') (Fp' + Gq')}; 
or substituting for il its value, this is 
= p" (EG - F 2 ) q' 2 - q" (EG - F 2 ) p'q’, =-q' (EG - F 2 ) (p'q" - p"q) ; 
wherefore 
Ep' + Fq'\' _ _1 
fii 
nfn, 
{M-q'(EG-F 2 )(p'q"-p"q% 
and the equation becomes 
Q. (E\p' 2 + 2F 1 p'q' + G.q' 2 ) -2M+ 2 q' (EG - F 2 ) (p'q" -p"q) = 0; 
whence we foresee that the whole equation must divide by q.