1874 
d 
k= : 
PQS cosa cos3 cosy 
and 
R ol Ze 
9 cosa cos cosy 
Likewise, if «,,8,,y, are the angles between the tangent in the 
point P (a, y,2) to CP (p,,q,,s,) and the axes of coordinates, and 
R, is the radius of curvature of this curve in this point, we find: 
d 
Rk, = > ——__ 
P,Q,S,coset,cosB, cosy, 
and therefore 
R |__| cosa,cosp, cosy, 
R,| | cosacosBcosy 
§ 13. The tangent in the point P, to the curve Cp (p,, q,, 5) 
admitting the director cosines 
pe, ’ q'Y ’ sz, 1 
this line is normal, in the case of rectangular axes, in P, to the 
quadrie of the pencil 
BE ER SIA onan vaker (ZN 
passing through P,. So the surfaces of this pencil (26) cut all the 
curves of the system C'(p,q,s) and consequently also all the sur- 
faces generated by curves of the system C'(p, q, s) under right angles. 
Moreover the pencil (26) cuts any surface generated by curves 
C(p, q, 8) according to the orthogonal trajectories of these curves. 
The surface Q,.-, being generated by curves of any system C’ (y), 
see § 7, we find the theorems: 
I. Any quadric of the net 
pet + gy? +set + A(p,e?+q,y?+s,27) =u. . . (27) 
cuts any surface Oc, under right angles. 
I]. The orthogonal trajectories of the curves C (A,) situated on Oo, 
are the intersections with surfaces of the pencil. 
pa? + gy + s2* + A, (pe? + ny? + 8,2") = u 
UI. Any curve of order four forming the base of a pencil of 
quadrics belonging to the net (27) cuts any surface Occ, under right 
angles. 
IV. Zn particular the orthogonal trajectories of the asymptotic curves 
of Occ, are determined by the intersection with the two pencils of 
quadrics | 
pat dgqy sz =u, 
Pe Hay +327 =H. 
