Laplace's equation for the potential <;6 becomes, from Equation [136g], 



4 - fi "9 



-i(r-i) 



di. 



d 



(1 -n 



<9m J 



= 0; [137r] 



{1 - ^,^) i^^ - 1) dc 



and the antisymmetric stream function ip, defined on the basis of a positive axis drawn toward 

 fi = 1, according to Equations [136k, 1], is related to 96 as follows: 



dil/ o dd) dill r, dd) 



—- = k(l - n^) —- , —-= - k iC^ - 1) -— . 



[137s, t] 

 Suppose now, that a solid ellipsoid of revolution is given with a surface defined by 



2 22 



X y + z 



1, a> b. 



[137u] 



9 n 1/2 / y 



Then its ellipticity is e = (a - b ) /a, so that b ^ a \/ 1 - e ; and for this ellipsoid 



a' = a, 6' = 6, ^ = ^» = a./k = 1/e. Thus k = ea = va - 6^, and on this ellipsoid 



2 1/2 

 a; = a/z, S" = 6 (1 - /i ) 



Five cases of the flow around the solid ellipsoid will be treated. In each ease ^ as 

 stated may be verified to satisfy Equation [137r], and 0, if it exists, to satisfy Equations 

 [137s, tj. The general case can be constructed by superposing flows of two or more of these 

 five types. 



Case 1. Translation of a Prolate Spheroid Parallel to its Axis of Symmetry at velocity 

 U, toward [i = 1, with g = in the fluid at infinity; see Figure 229. 



,1 <+ 1 



-A = -^ ff.k^Uie -l){l-fi^) ^ 



ffi = 



Co^-1 



1 ^=^0 + 1 



— In 



<o-l 



4-1 



1 - e' 



1 4 + n 

 - — In 



2 C-1 



1 1 + e 



— In 



1 - e 



[137v] 



[137w] 



[137x] 



348. 



