544 



Hence 



^ ^_^ *fy- 

 dx dy dydx ( 

 where 



V= {(?y-<2>-^} \(px-z)t qxs} - \(qy-z}s-pyt] \(px 



= *(* px - qy}(rt s~) . 



Hence _ 



cos v V 1 -f ff 2 + <f (zpxqyf 



But if OT be the perpendicular let fall from O on the tangent plane 

 at P, ___ 



zpxqy= V 1 +/ 2 -Hf .TZT, 

 and therefore 



COS z'< COS V . 



p 3 rts z 



But z3'=/3cosr. Also the quadratic determining the principal radii 

 of curvature at P is 



and therefore if v v v 2 denote the principal radii of curvature, 



Hence 

 and 



cos<)<j) 



COS i'2(f> l ~~CQSi 1 C([) l COS *'0' ~" COS *^ f . V^vJ 



which proves the proposition enunciated. 



In the particular case of an ellipsoid of revolution of which n is 

 the axial and m the equatorial semi-axis, compared with a sphere of 

 radius unity, hoth having their centres at O', one of the principal radii 

 of curvature is the normal of the elliptic section, which by the pro- 



perties of the ellipse is equal to m', m f denoting the semi-conjugate 



diameter ; and the other is the radius of curvature of the elliptic 



m 13 

 section, or . Also vr is the perpendicular let fall from the centre 



