Hence, using hyperbolic formulas in Section 32 and z = x \ iy, 



nail ^nax naU . 27ray 



sinh , = - sin 



26! ,2 2G ,2 



[90b, c] 



1 / 2nax 2nay 

 G = — I cosh - cos — 



2 \ .2 ,2 



'• = (« +y ) , 



[90d,e] 



? = 



,2 „2 



r'G 



[90f] 



These formulas represent a uniform stream flowing past two cylinders of radius a in 

 contact along a common generator, which passes through the origin. The fluid approaches at 

 velocity U toward negative x and hence perpendicularly to the plane through the axes of the 

 cylinders, as illustrated in Figure 146. 



Figure 146 - Flow past two similar cylinders 

 ,in contact along a common generator. 



Or, if a boundary is inserted along the ar-axis and only half of the diagram is used, the 

 flow is represented past a cylinder resting against a plane wall. Streamlines for the latter 

 case are shown in Figure 147. The excess of pressure above the pressure at infinity, for 

 steady motion, is shown in the figure as p - p^, along the positive a;-axis up to the origin, 

 and then around the right-hand half of the cylinder; the abscissa for the latter part of the 

 curve represents the angle 9, plotted toward the left. 



On the X-axis, tp = 0, r = - x, q = \u\, and 



2n^ a^ U 



a^ U ( 2na \ 

 I cosh -1 I 



x" \ ^ I 



n^a^U 



(na \ 



[90g] 



219 



