324 ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. [766 
The notation is that of the Memoir, “ Disquisitiones generales circa superficies 
curvas” (1827), Gauss, Werlce, t. ill.; see also my paper “On geodesic lines, in 
particular those of a quadric surface,” Proc. Lond. Math. Society, t. iv. (1872), 
pp. 191—211, [508]; and Salmon’s Solid Geometry, 3rd ed., 1874, pp. 251 et seq. 
The coordinates (x, y, z) of a point on the surface are taken to be functions of 
two independent parameters p, q; and we then write 
dx + ^d 2 x = adp + a'dq + \ {adp- + 2a' dpdq + a" dq 2 ), 
dy + \d 2 y — bdp + b'dq + \ (ftdp- + 2ft'dpdq + ft" dq 2 ), 
dz + %d 2 z = cdp + c'dq + (<ydp 2 + 2y'dpdq + y" dq 2 ) : 
E, F, G = a 2 + b 2 +c 2 , aa' + bb'+cc', a' 2 + b' 2 + c' 2 ; V 2 = EG — F 2 ; 
and therefore 
ds 2 = Edp 2 + 2 Fdpdq + Gdq 2 , 
where E, F, G are regarded as given functions of p and q. 
To determine a curve on the surface, we establish a relation between the two 
parameters p, q, or, what is the same thing, take p, q to be functions of a single 
parameter 0; and we write as usual p', p", q', etc., to denote the differential 
coefficients of p, q, etc., in regard to 9; we write also E lt E 2) etc., to denote the 
cLE dE 
differential coefficients -j—, , etc. In the first instance, 0 is taken to be an 
dp dq 
arbitrary parameter, but we afterwards take it to be the length s of the curve from 
a fixed point thereof. 
First formula for the radius of relative curvature. 
Consider any two curves touching at the point P, coordinates (x, y, z) which 
are regarded as given functions of (p, q); where (p, q) are for the one curve given 
functions, and for the other curve other given functions, of 9. 
The coordinates of a consecutive point for the one curve are then 
x+ dx \ d 2 x, y +dy + \d?y, z + dz + \d 2 z, 
where 
dp = p'd9 + \p"d9 2 , dq = <[d9 + \ q"d0 2 ; 
hence these coordinates are 
x + (ap + a'q') d9 + \ (ap- -¡- 2 a'p'q' + a"q 2 ) d0 2 + \ (ap" + a'q") d0 2 , 
and for the other curve they are in like manner 
x + (ap + a'q') d9 + \ (ap' 2 + 2a'p'q' + a"q' 2 ) d9 2 + \ (aP" + a'Q") d0 2 ,