766] ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. 
325 
the only difference being in the terms which contain the second differential coefficients, 
p", q" for the first curve, and P", Q" for the second curve. Hence the differences 
of the coordinates are 
£ [a (P" - P") + a' (q" - Q")} de\ i {b (p" - P") + V (q" - Q")} d6\ 
£{o(p"-P") + c' (q"-Q")}d0 2 , 
and consequently the distance QQ' of the two consecutive points Q, Q' is 
= l V(P, P, G\p" — P", q"-Q'y d&\ 
The squared arc PQ 2 is 
= (E, F, G\p', q'fdO 2 - 
and hence, if as before 2p . QQ' = PQ 2 , that is, - = 2QQ' -r- PQ 2 , then 
P 
1 f(E, F, G\p" - P", q" - Q'J 
P ~ (E, F, Gfp', qj 
the required formula for p. 
Second formula for the radius of relative curvature. 
We now take the variable 6 to be the length s of the curve measured from a 
fixed point thereof, so that p, p", etc. denote , etc. We have therefore 
1 ={E, F, G%p'> q') 2 , 
and the formula becomes 
- = f(E, F, G^p" - P ", q" - Q'J. 
P 
But, differentiating the above equation as regards the curve, we find 
0=2 (E, F, G\p', q'W> f) + (P> P> ¿IP'* 
where E, F, G are used to denote the complete differential coefficients E x p + Epf etc. 
And similarly, differentiating in regard to the tangent geodesic, we obtain 
0 = 2 (E, F, Glp\ q'QP", Q'') + (E, F, G^p', q') 2 -, 
and hence, taking the difference of the two equations, 
0 = (E, F, Gftp', q'W-P", f-Q")- 
Hence, in the equation for —, the function under the radical sign may be written 
P 
(E, F, G\ V ', qJ.(E, F, GW-?"’ r-Wy-m F,