18 



The corresponding formulas for line distributions and surface distributions are immediately 

 evident. 



THE FORCE F^ 



The same general procedure is followed for F^ as for Fj. We again consider S' to be 

 composed of a set of spheres surrounding the singularities, and define 



(i)=--^ I p[r{q ■ n) +^n] da 

 dt Js, 



[31] 



Fa = 2 F^(i) 



[32] 



If we m.ake the substitution 



[31] becomes 



i-Ji) = -p 



dt 



r = r. + R 



R(q . n)da + r. (q • n)d(j + <^nda 



[33] 



Remembering that these integrals are independent of R^, it can be seen from Equations 

 [23], [25], and [28] that 



R(q • n)da 



can involve only the coefficients a° , aj , ^. These are the strengths of doublets with their 

 axes respectively parallel to the x, y, z axes. The potentials of these doublets are 



a° cos d aj sin cos X 6j sin Q sin \ 



If we regard these coefficients as the components of a vector, this vector will have the direc- 

 tion of a single doublet equivalent to the three doublets, and its magnitude will be the strength 

 of this "resultant" doublet. We designate this vector by A, and call it the vector doublet 

 strength of the singularity. The potential and velocity field of a doublet in terms of its vector 

 strength may be written 



j.(L\ - A • n _ A • R 



[34] 



