Then, also, 



f(x) = 2r^E^{x) + 2r^Ri(x) + CV{x) + Q(x) [4?] 



It is seen that f(x) is a function of five variables, i.e., explicitly, 



r(x) = f(x,m; r^, r^ , C^) < 

 Now consider the equation 



f(x,m; r^, r, , Cp) = [44] 



For fixed x and m, it is seen from [43] that [44] is the equation of a plane 



in an (r , r , C ) rectangular coordinate system. This plane divides the 

 1 p _ 



(r , r^, C ) space into two parts, in one of which f(x) > 0, in the other 

 f(x) < 0. Hence Condition [4t ] is satisfied on only one side of the plane. 

 We may now consider x and m as parameters which define a two-parameter family 

 of planes in the (r , r , C ) space and an envelope surface. As x and m are 

 varied, the region of permissible values of r , r^ , and C becomes more re- 

 stricted as the successive planes intersect and reduce the space in which Con- 

 dition [4l ] is satisfied. Indeed, one can readily convince himself, by sketch- 

 ing an element of an uninflected curve and drawing a succession of tangent 

 lines to it (so that the curve element is the envelope of these tangent lines), 

 that the side of the envelope curve towards the center of curvature of the 

 element remains on the same side (plus or minus) of all the tangent lines to 

 the element. Similarly, in the three-dimensional (r , r^^, C ) space, the en- 

 velope surface delineates the region in which Condition [4l ] is satisfied. 

 Since the envelope surface may have inflection points and may Intersect Itself, 

 the permissible region may become further restricted. The equations of the 

 envelope surface, in parametric form, are 



f(x. m; r^. r^. Cp) = 



15=0 > [45] 



dm 



For each x and m Equations [45] are three linear equations which may be solved 

 simultaneously for r^, r^, C , giving a point in the (r^, r^, C ) space corres- 

 ponding to each pair of values (x, m) . 



