326 
ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. 
[766 
which is identically 
= (EG - F’) \p' (q" -Q')- 4 Cp" - F‘)Y. 
Hence, extracting the square root, and for VEG — F 2 writing V, we have 
\-7\p'(^-<r>-4(f -in), 
P 
or say 
\ = V(p'f-qp”)-V(p'Q"-q’P"), 
r 
which is the new formula for the radius of relative curvature. 
Formula for the radius of geodesic curvature. 
In the paper “On Geodesic Lines, etc.,” p. 195, [vol. viii. of this Collection, p. 160], 
writing EG — F 2 = V 2 , and P", Q" in place of p", q", the differential equation of the 
geodesic line is obtained in the form 
{Ep' + Fq') {(2F 1 - E 2 ) p' 2 + 2G lP 'q' + G 2 q' 2 } 
- (Fp + Gq') {E lP ' 2 + 2E,p'q + (2F 2 - G,) q' 2 } 
+ 2V 2 (p'Q"-q'P")=0; 
or, denoting by il the first two lines of this equation, we have 
V(p'Q"-q'P")=-^n. 
The foregoing equation gives therefore, for the radius of geodesic curvature, 
l p =V(p' q "-p"q') + tn, 
which is an expression depending only upon p', q', the first differential coefficients 
(common to the curve and geodesic), and on p", q", the second differential coefficients 
belonging to the curve. 
Observe that il is a cubic function of p', q': we have 
n = (S(, s, 6, $5/, qy, 
the values of the coefficients being 
21 = 2EF X - EE a - FE lt 
23 = 2EG 1 + 2FF 1 - 3FE, - GE X , 
6 = EGq + 3FG, - 2FF 2 - 2GE\, 
2) = FG 2 — 2GF 2 + GG X .