17 



LIMITATION OP RANGES OP PARAMETERS 



Not all combinations of the parameters r , r , C , and m give desir- 

 able forms. Por example, since for arbitrary values of the parameter, the 

 right member of [14] may be considered as an arbitrary polynomial of the 

 sixth degree, (subject to the restriction that y^ = -^ at x = m) it could very 

 well become negative (which would be meaningless) or assume several maxima 

 and minima and inflection points in the interval ^ x g 1 . Indeed maxima 

 greater than the desired one at x = m may occur. 



The term "desirable" remains to be defined, but for the development 

 of forms which may be considered as streamlined bodies, simple geometrical 

 restrictions may be imposed. The simplest conditions for ensuring a desirable 

 form appear to be that the polynomial remain positive in the range g x ^ i 

 and that the body have no inflection points. Since the form has a maximum of 

 y = -^ at X = m, and crosses the x-axis at x = and 1 , these conditions alone 

 preclude the occurrence of minima or of more than one maximum in the form of 

 the body. 



There may be occasions, however, where bodies with inflection points 

 near the tail are permitted or desired. Por example, such forms have been 

 proposed as laminar-flow forms. In this case a suitable condition is that the 

 sectional-area curve have no stationary value (maximum or minimum) other than 

 at X = m. 



The aforementioned conditions may be formulated mathematically and 

 employed to determine permissible ranges of the parameters. Pirst, let us 

 consider the maximum or minimum condition. 



MAXIMUM OR MINIMUM CONDITION 



Put y^ = f(x). The derivative f'(x) is known to be divisible by 

 (x - m) since f'(x) = when x = m. Removal of this factor leaves a polynom- 

 ial, ^(x) for example, which must not pass through zero over the interval in 

 order to avoid a minimum of f(x). Indeed, since f'(x) is positive near the 

 nose, when x - m is negative, it is seen that f(x) must be negative. Hence 

 the condition may be expressed in the form 



X - m 

 for all values of x in the range ^ x g 1 . Put 



?(^) = t44<o [^^] 



R„'(x) _ R/(x) ,, , ,, > 



