29 



Differentiating, 



dt dt ° dt 



dk' _dk^^ tZ" + a°^ 

 ° dt 



At time t this becomes 



dk' 

 dt 



dt 



[72] 



where v^- is the velocity of the singularity relative to the body. Equation [39] then becomes 



d^l dk 



F,ri; = -477, 



a>,-r,x.).r,:^."^] 



[73] 



POTENTIAL OF THE UNDISTURBED STREAM . <f>^ 



It has been seen that the coefficients o^, 5^ need be determined only for ^, the poten- 

 tial of the undisturbed stream.. In general, these coefficients can be found in terms of the po- 

 tential and its derivatives at point r^.. Since ^ is analytic in the neighborhood of r., it can be 

 expanded in a Taylor's series about r.. 



^(r) = <^(r.) + X(R . V )4>{r,) +^ (R ' ^^ )' ^^(r,) +|^(R •^)' <^(r,) + ••■■ 



where 



dx dy d 2 



= ff/cos 61 -^+ sin cos A ^ + sin d sin A. A\= R(n • V) 

 \ dx dy dz) 



Hence, the expansion can be written 



[74] 



Equating coefficients of /? " in this expansion and the expansion of the potential in terms of 

 spherical harmonics, we obtain the system of identities 



