In the plane containing the lamina but between it and the walls, = 0, t) = and 



dijj g\ cos y\ 

 sin rf/ = g sin y, u = = . [65g,h] 



dy |4T 



g^ sm^ y 



Without the walls, the formulas may represent a stream falling perpendicularly upon a 

 grating composed of such laminas lying in a common plane and spaced n apart; see Figure 98b. 



The similar transformation cosh w = g cosh s replaces the lamina between walls by an 

 opening of width 2 sin~^ 0-/g) in a transverse partition between the walls; see Figure 98c. 



Kinetic Energy of the Fluid 



If the lamina moves in translation parallel to the walls at unit velocity, with the fluid 

 at rest at infinity, the complex potential becomes tc - s, where w is still given by [65a]. Sub- 

 stituting w - 2 for u; in Equation [76a], and V = 1, the kinetic energy of the fluid is 



1 r -, 



^1 =2 P (^') ^ {w-z)d3, [65i] 



since for the lamina S = 0. Let the path of integration be displaced into a long rectangle with 

 sides on the walls and ends at ar = ± Z. This does not alter the value of the integral; see Sec- 

 tion 29. On either wall, dz =- x and w - z = (/) -x since ip = y; hence it is easily seen that 

 the contributions of the walls to the integral cancel each other. Over each end, ^ is practi- 

 cally constant and equal to its value at the corners, so that, when x = I and I is large, from 

 [65d], sinh being positive e*?" = ge", approximately and = a; + Ing; whereas when x =-l, 

 sinh (/) is negative, e~^ = g e~'' and ^ = a; - ln$'. Thus the integral over the two ends is 



77 77 



f(w-z)dB= / (Ing + itj/ - iy) idy + / {-Ing + iip - iy) idy = 2 f {\ng) idy = 2m\ng. 



77 



Hence 



7*1 = 77 p In y. 



Generalization 



The distance between the walls, or between the centers of the laminas in the grating, 

 may be changed from 77 to a, and also the velocity at infinity from unity to U, by substituting 

 in all formulas 772/a, 77 a:/a, ny/a, nw/aU, 7T<f>/alJ, nilz/aV, u/U for z, x, y, w, ^S, i/«, u, 

 respectively. Thereby all velocities are multiplied by U; the width of the lamina becomes 



L=-cos-l-; [65j] 



77 g 



and the kinetic energy of the fluid per unit length of the lamina is 



U7 



