'^ dx \ r^l dcj 3 



whence 



r = -7/ + 77, 



1 -2 &2 _ + _ [i21f] 



\ r3 r4 / 



A stagnation point Q occurs on the a;-axis where, to make (7=0, ar= 0, so that x = r = h. 



Clearly on the entire positive a;-axis i]j = h'^U and is constant. But ijj = h'^V also on a 

 surface of revolution S defined by the equation 



^2 = 262 /i-i\= 0^,2 (i_cos6i) [121g] 



where cos d - x/t. On this surface oT has a maximum value of 2 6 at = n-, or as a; -► - oo and 

 r -► |a;| . As a; increases algebraically, w decreases, and vanishes when 0=0. To find x at 

 this latter point, substitute t= y/x'^ +1^^ and rationalize, obtaining 



Z'* + (a;2-462)S2 ^ 4^2 (52.^2) ^ g [121h] 



AtS'= 0, a; = 6. Also, differentiating [121h], [4w^ + 2 (a;2-462)SJ] (dST/dx) + 2(^2-462) x = 0, 

 whence, as oT -> 0, doT/dx ■* - <^. Hence the surface S cuts the a;-axis perpendicularly at the 

 stagnation point or x - b. 



Thus the streamline for i// = 62^', approaching from both sides along the 2;-axis, divides 

 at Q into a sheaf of lines that extend off to infinity along S. The surface S divides space 

 into an exterior region occupied by fluid belonging to the incident stream and an interior 

 region occupied by fluid that has come from the source. 



If a solid boundary is introduced along S, no singularities occur outside it. flence 

 the formulas represent flow past a body of this shape, or a blunt-nosed cylinder of asymptotic 

 diameter 4 6. Its shape is fixed uniquely, since, if 6 is changed, Equation [121g] remains 

 satisfied when all coordinates are changed in proportion to b. 



In Figure 202 are shown some of the streamlines on a typical plane through the axis 

 of symmetry; the lines are equally spaced at infinity and thus differ by equal increments of 

 the quantity i/z/w'. The excess of the pressure p above the pressure p^ in the stream at 

 infinity, when the motion is steady, is also plotted, for points on S or on the a;-axis in .front 

 of it. On S, p ^ p^ at 2r ar = 62 or a; = 6/x/6"= 0.408b. 



To find the total force on the solid, which must be parallel to the axis by symm.etry, 

 select a narrow ring cut from its surface by two planes perpendicular to the axis, as 

 illustrated in Figure 200. The circumference of the ring is 2/707, hence its area is ^ircods, 



306 



