Within the spout, except near the entrance, is large and negative and q -1, approxi- 

 mately. The total volume of the inflow per second, per unit of length perpendicular to the 

 flow, is the value of A i// between the walls or 2 n. Along either wall q = |w| = (1 - e^)~^; 

 hence at the edge, where (^ = 0, q -* <x. Inside the spout, ^-»-ooasa;->-oo and q ■* 1, where- 

 as outside (^ -> + oo along any streamline and ^ -> 0. On the central or a;-axis, q = \u\ = (l + e^)~^. 



The more general transformation 



z ^c(gw + eS"") [62h] 



X = c[gcf> + eS0 cos igi,fi)], y = cigip + e^'^ sin {gifj)], 



where c and g are real constants and c > 0, represents a spout 2nc wide; for, when gifj =- n, 

 y -'^ nc. All velocities are changed from the values stated previously in the ratio \/cg. If 

 g < the fluid is issuing from the spout, but the flow pattern is the same. On the walls, 

 i/f = ± jt/g^ and the volume of outflow is 2n/g. 



(For notation and method; see Section 34; Reference 1, Article 66.) 



63. DIVERGING SPOUT 



2 = ^- — ^ (1 - e-""^) + e(i - ">"', < 7i< %; [63a] 



n 



—^ [1 _ e-"<?^ cos (mf,)] + e(l - "^^ cos [(1 - n)i/,], [63b] 



n 



= ^ e~"'^ sin(ni/,) +e(l-")0 sin [(1 - n)^,], [63c] 



— = (1 - n) (1 + e"^) e-""'. 

 dw 



This is a generalization of the preceding transformation, to which it reverts if n ^ 0. 



The streamline for if/ = is again the a;-axis. If (/< = ± 77, since cos (l-n.)n- = - cos nn, 

 sin {l-n)7T = sin nn, 



= - ( + e<^ J e""^ cos nn, 



n \ n J 



y = ± ( -^ + eA e""^ sin nn 



or, as may be verified by substituting the formula for x, 



142 



