GEOMETRY 57 



2.649 The direction cosines to the normal to the surface in the form 2.648 aje: 



i n 



' 



/a(/2, /3)V /a(/, /i)V , /a(/i, / 2 )\ 2 ] * 



VV5JK 5V UK 5"; *V(, t)/ 1 



2.650 If the equation of the surface is: 



-/(*, y), 



the equation of the tangent plane at #1, y\ t z\ is: 



2.651 The direction cosines of the normal to the surface in the form 2.650 are: 

 l, m ,n= 



fr/ \dy. 



2.652 The two principal radii of curvature of the surface in the form 2.650 

 are given by the two roots of: 



- 5 2 )p 2 - { (i + ?> - 2pqs + (i + p*)t}i + p2 + q* p + (i + f + q*)* = o, 



where 



2 2 " 



' 



df df 6 2 / a 2 /" 



' = J " 1 



2.653 If pi and p% are the two principal radii of curvature of a surface, and p 

 is the radius of curvature in a plane making an angle </> with the plane of pi, 



i _ cos 2 <fr sin 2 <t> 



P " Pi P2 



2.654 If p and p' are the radii of curvature in any two mutually perpendicular 

 planes, and pi and p 2 the two principal radii of curvature: 



i i = J 1 9 



P P' Pi P2* 



2.655 Gauss's measure of the curvature of a surface is: 



SPACE CURVES 



2.670 The equations of a space curve may be given in the forms: 



(a) Fi(x, y, z) = o, F t (x, y, z) = o. 



(b) * 



(c) y 



