expansion; the latter by means of a polynomial. In either case It is possible, 

 for the direct problem, to solve for the coefficients of the expansion for a 

 body of prescribed geometrical characteristics by means of sets of linear 

 equations. This is illustrated for the polynomial in the following section. 

 The direct application of the method of linear equations is tedious, however, 

 and another method — in which the equation of the sectional-area curve is given 

 directly as a linear combination of tabulated functions — is developed. As 

 will be shown, the determination of the latter functions is simplified in the 

 case of the polynomial form because of the property that its zeros appear as 

 factors. Furthermore, the polynomial form appears to be more suitable for the 

 purpose of computing pressure distributions. Because of its advantages, the 

 polynomial representation is used in the succeeding developments. 



EQUATIONS FOR POLYNOMIAL COEFFICIENTS 



The equation of a meridian section of a body of revolution will now 

 be expressed in terms of rectangular coordinates (X,Y) with the X-axis taken 

 along the axis of the body and the origin at one end (the nose) of the body. 

 Assuming a polynomial for the equation of the sectional-area curve, then 



ttY^ = A^X + A^X^ + . . . + A^X'^ [1 ] 



It will be convenient to operate with this equation in dimensionless 

 form. For this purpose put x = X/i , y = Y/d. Then Equation [1] may be writ- 

 ten as 



y^ = a^x + a^x^ + . . . + a^x" [2 ] 



where 



''s - "s „i2 



a= = A,-^, s = 1,2, ... n [3] 



Sketches of a sectional-area curve in dimensional and dimensionless forms are 

 shown in Figures 1 and 2. 



The coefficients aj^, a^, .... are to be determined in terms of pre- 

 scribed values of the geometrical parameters m, r .Tj,, and C . In the dimen- 

 sionless form the length and maximum diameter are unity so that X is elimi- 

 nated as a parameter. The length and diameter conditions are then that y = 

 when X = 1 , y = p- when x = ra, and gf = when x = ra. These respectively give 

 the equations: 



