17 



1. A singularity can be considered to be composed of a number of superimposed singulari- 

 ties, (ij^), (i^), (i^) "•, and the forces F^fz^ j, F ^(i^), ^ ^^(i^) "• determined independently. 

 Then 



2. Similarly, the potential excluding the singularity can be considered to be composed of 

 a number of superimposed potentials, and the force due to the interference of each of these 

 with the singularity can be determined separately and F .(i) found by addition. 



3. Consider the net force on the body due to the mutual interference of two of the singu- 

 larities within S. By 2, these forces can be determined without consideration of the effects 

 due to all other components of the flow. Instead of evaluating these forces separately over 

 the spheres iS^- and S-, let the integrals be taken over a larger sphere S^- with its center at 



fj. and R > |r. - r J. The combined potential may be expanded in a form such as Equation [27] 

 which will be convergent for R > |r . - r^l. However, since the combined potential must vanish 

 at infinity, all of the unbarred coefficients must be zero. Since the integrals will have pre- 

 cisely the same form as Equation [29], the components must be zero due to the bilinear nature 

 of Equation [30]. 



4. In evaluating Equation [30], the unbarred coefficients may be determined for ,p , the 

 potential of the undisturbed stream only, rather than the total potential excluding the singu- 

 larity at r^., since by 3 the net force due to the mutual interference of all the body generating 

 singularities is zero. 



5. In the case of continuous distributions, we may suppose the region over which the 

 singularities are distributed to be subdivided into small elements. The net potential A -cj), 

 of the portion of distribution within the element A^ t, containing the point r • can be written 



ix> n 



\'f'b= ^ ^/^^"^^^P^(f^)(«^cos s\ + ^^ sin sX) A.T 



n=o s-o 



which converges for all R. greater than the maximum distance from the point r. to the bounds 

 of Aj-T. This has the form of an isolated singularity at r^. Hence Equation [20] can be written 



^y=2J^,(i)«'K' <.^:) A, 



If the number of elements is increased indefinitely, the dimensions of each approaching zero, 



s 



then the coefficients ce^ , /5^ will in general approach limits, and the sum becomes an inte- 

 gral. 



■ =/^. 



(i)«> K><,0t)dr [20'] 



