satisfying given conditions. 169 



Now when P and Q are corresponding points on the two 

 curves, the tangents at P and Q will be perpendicular, and there- 

 fore the normal to the surface at R, being perpendicular to the 

 tangent and binormal at P, will be parallel to the principal 

 normals of the three curves. 



Thus the third curve is a geodesic on the surface ; moreover, 

 since the tangents at P and Q are perpendicular, and also parallel 

 to conjugate lines at R, they are the directions of the lines of 

 curvature at R ; but the tangent to this geodesic makes a con- 

 stant angle with these two tangents: accordingly we have a 

 geodesic on the surface which cuts the lines of curvature at a 

 constant angle. 



Further, if R 1 and R 2 are the principal radii of curvatures of 

 the surface at any point, we have along this geodesic 



1 cos 2 -v/r sin 2 -v^r 



p~ = ~rT + ~W' 



1 • r i I 1 1 



- = Sin vr COS vr < -jy — -fr 



<T [-til R< 



thus we can find the relation between R x and R 2 which is satisfied 

 along this geodesic. 



Returning to the determination of special curves, the following 

 are convenient forms in certain cases : 



CC — Cv J Ju^CVSq , 



x = af(z dy - y dz ), 

 x = ajoo tan %cfe , 



x' = a Jx t) ds cos t\t + (2 dy — y dz ) sin -yfr, 

 x' = a/vtan ^ {x ds + z dy — y dz n ], 



x = afx de . 



As an example of the method, suppose we take as our in- 

 dicatrix a helix on a sphere: its equations may be written 



n a . ( . a\ . „a . / aQ Q a 

 cc = cos- ~ sin I 20 sin- - j — sin- ^sm I 20 cos- - 



y n = cos- ~ cos I 20 sm- ^ I — sin- - cos I 20 cos- ■= 

 z = — sin a . cos (0 cos a). 



