MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



= o, 



where: 



2.644 The coordinates of each center of curvature are: 

 i ._odF o dF 



dF 



2.645 The envelope of a family of surfaces: 

 i. F(x, y } z, a) = o 



is found by eliminating a between (i) and 



2. 



da 



O. 



2.646 The characteristic of a surface is a curve defined by the two equations 

 (i) and (2) in 2.645. 



2.647 The envelope of a family of surfaces with two variable parameters, 

 a, /3, is obtained by eliminating a and j8 between: 



F(x, y, z, a, /3) = o. 

 dF 



dF 



a/3 



= o. 



2.648 The equations of a surface may be given in the parametric form: 



x =/i(w, u), y =/*(, ), z =Mu, ).- 



The equation of a tangent plane at %i, y ly z\ is: 



where 



dfr dfr 

 du dv 



, etc. See 1.370. 



