NO. 4 WIND TUNNEL EXPERIMENTS IN AERODYNAMICS 8l 



By the hypothesis this velocity is —u sin 6, where u is the velocity of 

 the stream. Hence, when ^ is large, we must have for all values of 

 fi approximately 



-^^- = ^ , = — Jf, mdependent of u. (7) 



It follows that M cannot be of higher order than i, and the only possi- 

 bilities for n are o and i. The solution for cf> must therefore be of the 



form 



<^ = Z, + /.Z„ (8) 



where Z^ and Z^ are solutions of (6) for n = o and i respectively. 

 Moreover, for the radial velocity in distant regions we have 



deb dZ,, dZ, „ .. 



- J^=~ ^^l -^^ ^ =ucose = u^. (9) 



We have next to express the condition that flow along the ellipsoid 



shall be tangential, i. c, that the normal derivative -, shall vanish. 

 ^ ' dn 



The normals to the elliptical section of the ellipsoid are the hyper- 

 bolas along which /x is constant. The normal dn is 



dn=\/dx' + dr~ci^p."-+{i-l.-') ^^^^^\it=c^l^^^dl 



The particular ellipse which is the profile of the disk is determined 

 by some value Co o^ C- Then 



As this equation holds for every /x, we have 



f.) =o.('f.) =0. (10) 



dC I f=fo \dt I f.fo 



The equations (7), (9), (10) should suffice to determine what 

 values of Zq and Z^ are needed in (8) to represent the flow. 



The general solution of (6) for « = o may be obtained at once by 

 integration, 



Z^^C^^C, tan-^C. (II) 



The general solution for n=i may be obtained from the above- 

 mentioned particular solution t, by the substitution Z^^Z't,, which 

 gives 



Z, = /Ci (Ctan-^C+i)+i^,C. (12) 



From (10), (11), (12) we find: 



^=0, /v.(.a.r.<,+ ^^f--)+if, = o. 

 6 



