Also, the elements of distance in the coordinate directions, calculated as ds = {dx^ + dy^) 

 with either drj = or d^ = 0, 



dst= cGd^, ds = cGdri, [82q,r] 



G = (sinh^ ^ + sin^ rj) ' = — (cosh 2 ^ - cos 2 jj) . [82s] 



by hyperbolic formulas in Section 32. Hence the components of velocity q ^ and q in the 

 coordinate directions are, from Equation [6f], in which ^ denotes the velocity potential, 



1 (90 1 dd> 



^^ cG d^ ^"1 cG dr, 



In applying these results it may be more convenient to substitute, in place of f, the 

 semiaxes a, h, of the corresponding ellipse, always taken positive. Then, if always ^^0, 

 from Equations [82c, d] and [82i,j] 



X = a' cos T), y = 6' sin 77, ^ = In [(a'+ b')/c] [82v, w,x] 



For any point (x, y), the value of a' can be found by adding distances from the foci and 

 dividing by 2; then 



b = (a - c ) , tan rj = a' y/b'x. 



The components of velocity, q^ along the tangent to the ellipse in the direction of 

 increasing 17, and q^ along the outward normal, are then, from Equations [82t, u] and [82j] 



9„ = ?f=- — — , ?,= 9^=- — — , A'=(6'^ + c2 sin%) [82y, z,a'] 



The components q^ and q^ make angles d^, 0^ with the aj-axis where, from Equation [82o,p], 



(9„ = ^^= tan~^ ( -7 tan tA , (9^ = 6 =- tan~^ i—j cot 7/] . [826' c'] 



Geometrically, the transformation from z to ^ maps the entire s-plane continuously onto 

 the positive half, and again onvthe negative half of a strip on the ^-plane parallel to the ^-axis 

 and extending from 77 = to r; = %n, as may be verified by consideration of the displacements 

 on the ^-plane that are required to reach all parts of the 2-plane. The mapping is then repeated 

 in this manner upon each successive parallel strip of width 2 77. 



195 



