58 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2.671 The direction cosines of the tangent to a space curve in the form (a) are: 



dF l dF 2 dF l dF 2 



dy dz dz dy 



- ~~ 



m = 



F^dFz dFi dF 2 

 dz dx dx dz 



dx dy dy dx 



n = = ^ 1 



where T is the positive root of: 



,3 77 .3 77 5i 77 .3 77 J3 77 ^77^2 



O/^ 1 Ox 9 OjP 1 C/x 2 0./1 C/x^9 I 



171J7 + l^l^ + 1^15j 



2.672 The direction cosines of the tangent to a space curve in the form (b) are: 



x', y', z' 



'' 



where the accents denote differentials with respect to /. 



2.673 If s, the length of arc measured from a fixed point on the curve is the 

 parameter, t: 



, dx dy dz 



/, m, n = > -j-> -r- 

 ds ds ds 



2.674 The principal radius of curvature of a space curve in the form (b) is: 



__ (* /2 + y 2 + 2 / *)t _ 



P ~ {(y'z" - z'y"? + (z'x" - x'z"? + (x'y" - y'x")*}* 



where the double accents denote second differentials with respect to /, and 5, 

 the length of arc, is a function of /. 



2.675 When t = s: 



2.676 The direction cosines of the principal normal to the space curve in the 

 form (b) are: 



v _ z'(z'x" - x'z") - y'(x'y" - y'x"} ^ 



f x '(x'y" - y'x") - z f (y'z" - z'y") 

 m - L 



