Figure 89 — Cylinder with semi-infinite 



cross-section S, obtained from a 



source-sheet in a stream. 



See Section 58. 



^U l-1 + ffln-^y v^gV(e-d^), 



[58d,e] 



where 



r=fx2+y2^^^ r^Ux + cf+y^]"^, = tan"! -, 9,=tan-l— , 



and d and 0. may be assumed to lie between -n and n and to have the sign of y; see 

 Figure 89. 



A stagnation point Q occurs on the ar-axis, which represents a plane of symmetry, at 

 X = Xq> where, to make u - 0, since r = x, r^ = x + c, 



Xq = c(el/g -l)-^ 



[58f] 



On the positive a;-axis, 0=6^=0 and i^ = 0, also, if/ = on the dividing surface S 

 defined by 



y+ g\x(d-d^)-cdj^ -y\n -^1 = 



[58g] 



By expanding all terms in powers of y it can be shown that S crosses the a;-axis perpendicu- 

 larly at Q. Since v has the same sign as y, the surface S, being a stream surface, must be 

 broadest at a; -» - oo. At large distances from the origin, - 0^ -► c sin &/r and Infr^/fj -♦ 0, 

 also, X -* r cos 6. Hence, Equation [58g] becomes, in the limit, 



y + g (c sin 6 cos 6 - cd) - 



Thus as a; -> - oo, 3/ remains finite; hence -* n and y -> cgn. Hence, toward negative x, S 

 becomes asymptotic to a cylinder of radius 



R = n eg 



[58h] 



132 



