v^gVyi , [54e] 



(1-1), 



[54f] 



Singularities occur at s = ^ o; and stagnation points Q ., Q„ occur where dw/dz = or 

 2 ^ X = — I and 



I = y/a^TVag [54g] 



On the a;-axis where x > a, r. = x - a, r^ - x + a, and 5' = |u| where 



u^U 1-1+ _?^1_) [54h] 



\ a;2-a2/ 



Thus the streamline for i/» = follows the x - axis from + » to $,, where it is joined by an- 

 other branch coming from the source; then it divides and proceeds along the two halves of the 

 curve S that is defined by 



2ay y 

 = tan ^ [541] 



From Q^, one branch of this streamline proceeds to the sink, the other follows the x - axis to 

 - 00. The surface S divides the fluid into that which is coming from infinity and that which is 

 on its way from the source to the sink. 



The curve S is symmetrical about both axes, and is called a Rankine oval. Since an 

 angle in radians and its tangent are nearly equal when the angle is small, the symbol tan can 

 be omitted in [54i] when y is small; then, after canceling y, it becomes clear that the curve 

 crosses the x - axis at the stagnation points. It is broadest in the middle, where x = 0. Its 

 half-width h can be found by putting y = k and a; = in [54i] and solving the resulting quad- 

 ratic for h; the result can be written 



a h 



-=tan^— [54]] 



h 2g 



On the middle circumference of S, q = \u\ and 



u = -V fl + -Jft-\ [54k] 



\ a2 + ^2 y 



125 



