19 



Because of the graphical difficulty of presenting a surface in three 

 dimensions the foregoing analysis will be Illustrated in a two-dimensional 

 case. We will suppose that both m and C are prescribed. Then, considering 

 X as a parameter, Equation [44] represents a one-parameter family of straight 

 lines in an (r , r ^^ ) plane. The envelope of this family of lines has the 

 equations 



f(x; r^, rj = 



ax 



For each x, Equations [46] are linear equations which may be solved simultane- 

 ously for r and r , giving a point in the (r , r ) plane. By computing the 

 pairs of values of (r , r^) for a sufficient number of values of x in the 

 range ^ x ^ 1, the envelope curve of r^ against r may be graphed. Figure 

 4 shows two envelope curves of r^^ against r for m = 0.40 and C = 0.55 and 

 m = 0.40 and C =0.65. The corresponding values of x are marked along the 

 curves. Proceeding along the curve in the direction of increasing x, the neg- 

 ative or permissible side of the tangent line at each x (representing f (x) = 0) 

 is on the right, so that the right side of the envelope curve is the permissi- 

 ble side. Consider the curve for C = O.65. For a point (r^, r^) to be per- 

 missible it must be on the right side of all possible tangent lines that can be 

 drawn to the envelope curve for ^ x ^ 1 . It is seen in Figure 4 that tan- 

 gent lines for x > 0.68 begin to eliminate regions which previously were on 

 the right side of the envelope curve. Furthermore only points (r , r^^ ) in the 

 first quadrant need be considered since negative radii are meaningless. In 

 this way the permissible regions for r , r^, bounded by segments of the r and 

 r axes and by arcs of the envelope curves, shown shaded in Figure 4, are 

 obtained. 



INFLECTION POINT CONDITION 



If the body Is to have no inflection points, its slope must be mono- 

 tonlcally decreasing as x increases from to 1, i.e., 



-^<0 [47] 



dx^ 



for all values of x in the range ^ x g 1 . Again put y^ = f(x). Then, dif- 

 ferentiating successively with respect to x, we get 



