For the radius of torsion o, of the curve Cp(p,, gs 5) in the 
same point we get 
eye QY =) 
1 nj 2 + ’ 
is B: 19,5. a ¥ od 
and, as P,=—P, Q,=—Q, S,=—S (see § 2, equation 11), 
Lel =le, |D 
Of the screws osculating the asymptotic lines of the surface O,., 
in any point the cne is righthanded, the other lefthanded, as the 
determinant 
LYz 
| a A 2 oP ee a) 
assumes opposite signs for the two asymptotic lines. 
Let X, Y, Z represent the director cosines of the binormal and 
d the distance of the origin to the osculating plane in the point 
(zy, 2); then we ca find : 
2 S P 
Lp ree ES er 
2 , & y 
= d _ ayz PQS 
EE Ose Ce ee 
Let Ag be the angle between the tangents to a curve of the 
system Cp,g,s) in the points corresponding to the values ¢ and 
t+ At; then we have: 
5 SZ 
PQs UYZ 
dp 
de t(p?2? 7 gy? + = 
by means of which we find for the radius of curvature Pf: 
3 
do eA (pet + g?y’? + as 
dp Q? 
pgs cy? (S Vs En =) 
or, if a, 8, and y are the angles eee i tangent and the axes 
of coordinates 
R= 
> 
1 atas 
Fp mos cos Bow y [5 He tel 
So we get: 
1) PAScAL, Rep. di Mat. Sup. Cap. 16; § 9. 
