14 



and 



y = 



_ 7(1 - 4m + 5m^) 

 2m^(2 - 7m + 7m^) 



10 - 28m + 21m^ 

 2(1 - m)^(2 - 7m + 7m^) 



7(2 - 6m + 5m^) 



2(1 - m)^(2 - 7m + 7m^) 



105 



2(2 - 7m + 7m ) 



3(2 - 8m + 7m'') 

 4m^(l - m)^2 - 7m + 7m^) 



5, =-■ 



2 - 4m - 14m^ + 21m'' 

 2m^l - m)^(2 - 7m + 7m^) 



7(1 - 5m + 5m ^) 

 4m'(l - m)'(2 - 7m + 7m^) 



[40] 



The coefficients a through 6^ are tabulated for values of m from 

 0.10 to 0.50, in intervals of 0.02, in Appendix 1. Appendix 2 contains tables 

 of the polynomials Rq(x), R (x), P(x), and Q(x) for values of x from to 1 , 

 in intervals of 0.02, and for the same values of m. Graphs of these functions, 

 for selected values of m, are shown in Appendix 3- 



The numerical example in Table 1 illustrates how the tables of Ap- 

 pendix 2 can be used to calculate the offsets of a given form. The figures 

 apply to a body whose geometric parameters are r = O.5O, r^ = 0.10, C = O.65, 

 and m = O.UO. The calculations for y^ and y, shown in Table 1, are based di- 

 rectly upon the tables for Rq(x), R^(x), P(x) , and Q(x) corresponding to 

 m = O.UO. The resulting body and sectional-area curves, in dimenslonless form, 

 are shown in Figure 3 . 



The graphs of the basic polynomials given in Appendix 3 are useful 

 in that they provide a visual means of showing how each geometrical parameter 

 affects the shape of the body. Thus, if a body with certain prescribed param- 

 eters is not suitable for the intended purpose, the graphs will indicate the 

 parameter changes that are necessary to produce a desired volumetric distribu- 

 tion or contour. 



