766 Dr. J. R. Wilton on the Solution of 



case of the' elliptic cylinder in a steady stream. Consider, 

 however, the stream function 



^ = (A + B)£-^1 [^ sec v +'-¥-cosQC7))dri 



r} — i% 



. . ™ „ Koc , , /sinh f \ By , , /sinh A 



= (A + B)f~ — tan" 1 ! 2 —-/tan M-, — ^ ). 



v ^ a \ cos n ) b \ sin 77 J 



We obtain the velocity at any point from the formulse 



d? ay + 0*7 d#' o? o# ~ 0*7 dy* 



And we have immediately 



O^ , n Ac f . , , ,, , , /sinh A cosh £ cosh (X + £) ) 



^ =A + B < sinh (A +£) cos 77 tan" 1 k )+ 1 , A i 2 w 2\ C 



of a ( w \ cos 77/ 1 -f (sinh 2 f /cos 2 77) J 



JBc T , ^ , -.v . , ,/sinh £\ cosh £ sinh (\ + f) 1 



— r < cosh (A,-r£)sini7tan — = — 3 -f - — , , 2 «., • 2 < f, 



I * \ sm 97 / 1 -h (sinh 2 f /.-m 2 77) J 



O^ Ac , , . C ,/sinhA sinh f cos 77 1 



^-?- = — cosh(X + f) sm<n< tan H ?)— b — ^- > 



0^7 « v *' ' ( \ cos 77/ smrr £ + cos J 77 J 



Be •' u /a . « K -i/ sl ' nn f\ sinh f sin 77 "\ . 



— -r smh(X + f) cos 77 < tan * — — M — . U2 «. , . 2 f 

 6 v w /^ \ sin 77/ smb. 2 £ + sin 2 77 J 



♦ At infinity we have 



O^ 7T(3 A ,,/A B . \ 



^r = -7- <? +f ( — sm 77— y- cos ?; I. 



And therefore 



7T B _ 7T A 



2T' "~2a 



Further, on the ellipse f=0 we have u=v=Q, provided 

 that neither cos 77 nor sin 77 vanishes. 



In the neighbourhood of the extremities of the minor axis 

 put 



sinh f = tan \yu cos 77. 

 Then, at these points, 



where — ir < fi -< ir, but jjl is otherwise arbitrary. 



