21 



2yy' = f'(x) [48] 



2y'^ + 2yy" = f"(x) [49] 



or, eliminating y' between [48] and [49], we obtain 



„ ^ 2ff' - f'^ [^ 



We must also suppose that y^ > 0, or 



f(x) > f or < X < 1 [51] 



Then also, from [47] and [50], 



2ff" - f '^ < [52] 



Both Conditions [51 ] and [52] can be shown to determine envelope surfaces 

 bounding regions in an (r , r^ , C ) space, and the region which is common to 

 both is the permissible one. 



In cases where all the parameters but one are prescribed and it is 

 desired to determine the permissible range of variation of that parameter, a 

 procedure somewhat simpler than the envelope method may be employed. For ex- 

 ample, suppose r , r and m are prescribed and the limits of permissible vari- 

 ation of C are to be found. Then the equation f(x) = may be solved for C 

 as a function of x and plotted against x. The resulting curve is the boundary 

 separating positive from negative values of f (x) . A value of C for which 

 Condition [51] is not satisfied for any x must be discarded. Hence the per- 

 missible range of C would be defined by the ordinates of the horizontal tan- 

 gent lines, at the maxima and minima of the curve of C against x, which do 

 not otherwise intersect the curve. Similarly the equation 2ff" - f'^ = can 

 be solved for C plotted as a function of x, and the ordinates of these tan- 

 gent lines, at the maxima and minima, which do not again intersect the curve, 

 obtained. The range of C which is common to the ranges satisfying [51] and 

 [52] separately then gives the desired range which satisfies the conditions 

 simultaneously. This procedure will be illustrated in a later section in the 

 case of a seventh-degree polynomial, where the parameters m, r , r^^ , and C 

 are prescribed, and a new parameter a„, to be defined, is varied. 



