Smith 



R P(X.Y,Z) 



Fig. 2 - Flow about a three -dimensional body 



Representation of the Solution by a Surface- Source Distribution 



In the present method, the solution is expressed as the potential of a source 

 distribution over the body surface. The potential at a point P due to a point 

 source of unit strength at q is l/r(P,q), where r(P,q) is the distance between 

 two points (Fig. 2). Accordingly, the potential at a point p with coordinates x,y,z 

 due to a source distribution cr over the surface s is 



s 



where q is a point on the surface S, and dS is an elemental surface area as 

 shown in Fig. 2. Reference 2 has shown that the function satisfying Eqs. (1), (2), 

 and (3) can indeed be represented in the form given Eq. (4). The function (p as 

 given by Eq. (4) satisfies Eqs. (1) and (2) identically for any function a. This is 

 true simply because the function l/r(P,q) satisfies these conditions. The function 

 CT is determined from the boundary condition on S, Eq. (3). Applying Eq. (3) re- 

 quires the evaluation of the limit of the normal derivative Eq. (4) as the field 

 point P approaches a point p on the surface s. The derivatives of l/r(P,q) now 

 become singular as P approaches p, and care is required in evaluating the limit. 



The limiting process is discussed in detail in Ref. 2. The results are stated 

 here without proof. The limit of the normal derivative of the integral of Eq. (4) 

 consists of two terms. One is the expected term, which is the integral of the 

 normal derivative of the integrand of Eq. (4) evaluated on the surface, i.e., P = p. 

 This integral is an ordinary integral, not a principal value, because its integrand 

 is integrable. The other term is something of a surprise. It is a "local effect" 

 term that expresses the fact that an infinitesimal neighborhood of the point p has 

 a finite contribution to the normal derivative there. As is shown in Ref. 2, the 

 "local effect" term is -27TC7(p), Finally, the result of applying Eq, (3) to cp as 

 given by Eq. (4) is 



320 



