328 
ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. 
[766 
say this is 
and the formula thus is 
= p'q' (Lp' + Mq') ; 
-*ir “ npY-p"q') + ^p'q'(Lp‘ + M,'). 
Taking cf), 6 to be the inclination of the curve to the curves q = const., p — const., 
respectively, and w (= $ + 6) the inclination of these two curves to each other, then 
cos 
»- J y +fl y. cos 
VG s!E 
• , Vp' . a Vq' 
Y ^G sJE- 
COS ft) = 
Sin ft) = 
F 
EG ’ 
hence 
sm ^ = p a/E, ^ = q' VG, and the formula may also be written 
sin o) 
sm 03 
= V(p'q"-pY) + êpY (Lp' + Mq'). 
p sm 03 U sm co o 1 r 1 x r 2 
The Orthotomic Case F = 0, or (is 2 = Edp- + (x<iç 2 . 
The formula becomes in this case much more simple. We have 
1 = Ep 2 + Gq 2 , V = Vi?6r, to = 90°, sin 6 = cos </> ; 
and the term Lp' + Mq' becomes = EG — EG, if, as before, E, G denote the complete 
differential coefficients E x p' + E 2 q' and Gqp + G 2 q'. The formula then is 
i -= V(p'q" -p"f) + i(JSÔ -ÈG), 
where the values 1 and f are now r = J ^ 
R 
moreover cb = tan -1 ~ , and thence 
q \/G 
-№ 
GTJË anc ^ F\JG ’ res P ective1 ^ But we h ave 
<*>' = S' VG (/' -JE+^^-p' V-E (s" VG + Ì0) 
= - V(pY-p"q') - ipY (EG - EG) ; 
or the formula finally is 
1 cos 0 sin 0 
+ (f)' = 0, 
p R S 
which is Liouville’s formula referred to at the beginning of the present paper. It 
will be recollected that <// is the differential coefficient with respect to the arc s 
of the curve.