If nB/aU > 1, S consists of two undulating surfaces enclosing-the plane y = 0. This 

 appears from Equation [47k], which cannot be solved for r^ if nB/aU > 1, so that S cannot cut 

 the a?-axis. 



If nB/aV < \, S breaks up into a set of similar cylindrical surfaces with axes at 



a = 0, - a, - 2a The semidiameters of each cylinder in the x and y directions, found as 



in Section 46, are r and r, where 



«i"^=[/^ ,r2-_coth_ [^71^^!^ 



Stagnation lines occur on the cylinders where y = 0. 



The formulas will then represent streaming flow past a grating whose bars have the 

 contours of the cylinders. If the diameter of the bars is smaller than the spacing between 

 them, they are nearly circular in section, the y-diameter exceeding the aj-diameter by less than 

 2 percent. Streamlines for such a case are shown in Figure 75; only half of the symmetrical 

 diagram is illustrated. If the diameter is small, r^ = r = {aB/nlJ)^^^ , nearly. 



(For notation and method, see Section 34.) 



48. ALTERNATING LINE DIPOLES; DIPOLE MIDWAY BETWEEN 

 WALLS, WITH PERPENDICULAR AXIS 



'(sinh^) 



w = i fi/fsinh — 1 [48a] 



B nx ny B ttX try r , -, 



4> - — cosh — sin — , d/ = — sinh — cos — [48b, c] 



G a a G a a 



ffS! . -^ TV s I , 2?ra; 27ry\ 



G = sinh"' — + sin"' — =- cosh cos 1 [48d] 



a a 2 \ o a I 



ttB I ^^nx ny\ . ^ ttX _ ny ^^^ ^ 



u = — r (cosh^ — + COS'' — sinh — sin — l48e] 



aG'^X a a I a a 



776 // , , rrar . , n-y \ nX ny , »^ 



V = =r sinh'^ sin-'^^^ I cosh — cos — [48r] 



aff-'V a a I a a 



Near (0,0), tt -» ia 6/772, representing a line dipole of moment aB/n at the origin with 

 its axis toward positive y (if S > 0). See Section 46. The entire field represents a row of such 

 dipoles spaced a apart along the y-axis and with their axes directed alternately toward positive 

 and toward negative y. 



In the conjugate flow the dipole axes are parallel to the a;-axis. 



114 



