Rj^ is the radius of curvature at the tail. 



V is the volume. 



Other characteristics, such as the surface area, the position of the center of 

 gravity, the radii of gyration, etc., are of Interest for various purposes. 

 These are considered as derived quantities in the present report, and are in- 

 cluded in Appendix 5- 



It is convenient to employ the following dlmensionless combinations 

 of the primary geometric quantities: 



, X R I Rd 



X = j-, m = -f , r = -^, r = 4-' 



d' I ' ^2' 1 ^2 



and the prismatic coefficient C = 



4V 



P TTd^l 



The question as to how well the foregoing parameters define the 

 shape of a body is discussed in a subsequent section. 



CHOICE OP MATHEiVIATICAL FORM 



For both mathematical and physical reasons the development has been 

 based on the sectional-area curve of a form, rather than a meridian section 

 of the form itself. Thus it will be shown that the slopes of the sectional- 

 area curve at the ends of a body are proportional to the radii of curvature 

 at the ends, a relation which greatly simplifies the determination of the 

 equation for a body. The physical reason is that the sectional-area curve Is 

 proportional to an axial doublet distribution which, to a good approximation, 

 generates the desired body In a uniform stream.^ Consequently it is desirable 

 to have simple mathematical expressions for the sectional-area curve of a body 

 for the purpose of computing the potential-flow field about it, and Its pres- 

 sure distribution. 



The question remains as to the most convenient mathematical form in 

 which the equation of a sectional-area curve can be expressed. Let us con- 

 sider for a moment the converse of the present problem; i.e., the determina- 

 tion of the geometrical characteristics of a given body, rather than the de- 

 velopment of an equation for a body of given characteristics. The geometrical 

 characteristics can be computed directly from the equation for a body. To ob- 

 tain its equation, a given body may be curve-fitted with any complete set of 

 orthogonal functions, each of which can-give a "best" fit in the least-square 

 sense.® Practically, however, It is convenient to employ, for this purpose, 

 either the trigonometric functions or Legendre polynomials. The former fit 

 the equation of given form by means of a finite number of terms of the Fourier 



