cylindec, or the line the lamina, the outer curves then representing streamlines. For this case 

 it is most convenient to keep (f> > 0, hence ifj' < 0. Then from [61b, c] it is easily verified that, 

 as ^'or i/f increases continuously from to 2/7, the point {x,y) passes once around the cylinder. 

 Thus the many-valuedness of (f> 'implies the existence of circulation around the cylinder of 

 magnitude 27t. 



On an elliptical cylinder defined by ^'= -<^ = -0,, with major semiaxis a, = c cosh 

 4>y, from [61d,f,b], after inserting sin^ i/r = 1 - cos^ i/y, 



1 / , x^ \ 1 /°l "^ \ 



?=' cosh2<;6 =- — _— [6ip] 



"^ \ c2 cosh2 0^/ C \ ^2 ^2/ 



At large distances the elliptical streamlines approximate circles, and, since sinh <^ 

 becomes large and nearly equal to cosh 0, it is readily seen with the use of [61e] that 



9 = 



approximately. Thus the flow approximates that of a line vortex (Section 40) at the origin. 



If 0'and i/r'are replaced in these last formulas by cf)'/k, ip'/k, then all velocities are 

 multiplied by k and the magnitude of the circulation becomes 27Tk. 



The variables and i/r, defined in terms of x and y by [61b, c], can be used as coordi- 

 nates on the a;y-plane; this use, and the geometrical properties of the transformation, are dis- 

 cussed in Section 82. 



Among other possible forms, 3 = ic cosh w gives the same field of flow rotated through 

 90 deg, with the foci at (0, -c); z = c cos w or 2 = c cosh (iw) is the conjugate transformation, 

 in which ^S, (/> are replaced by t/f, -(^; and 2 = c sinh m gives the original field rotated through 

 90 deg and with if/ increased by rr/2, to which the conjugate transformation is 2 = ic sin w. 



(For notation and method; see Section 34; Reference 1, Article 66; Reference 2, Sec- 

 tions 6.10, 6.30.) 



62. STRAIGHT SPOUT 



3 = w + e"'. [62a] 



Since 



e"^ = e9 + "A = e<P (cos if/ + i sin xp), 2 - x + iy, and - u + iw = dw/d2, 



X = 4^ + e^ cos i//, y = ip + e4> sin if/, _ [62b, c] 



140 



