92 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



A /a = - ua. Hence the motion produced in an infinite mass of 

 liquid by a circular cylinder moving through it with velocity u 

 perpendicular to its length, is given by 



11 r/ 2 



* = - sintf (3), 



which agrees with Art. 68. 



3. Let us introduce the elliptic co-ordinates f, rj, connected 

 with x, y by the relation 



x + iy = c cosh (f + iy) (4), 



or X G cosh f cos rj, 



y = c sinh f sin rj j 

 (cf. Art. 66), where f may be supposed to range from to oo , and 

 rj from to 2-7T. If we now put 



+ ty = (V-tf+*,) ........................ (6), 



where G is some real constant, we have 



i|r = - Ce~t sin rj ........................ (7), 



so that (1) becomes 



Ce~% sin 77 = uc sinh { sin T; 4- const. 



In this system of curves is included the ellipse whose parameter 

 fo is determined by 



If a, b be the semi-axes of this ellipse we have 



, 6 = csinhf , 



xl. r 7. 



so that (7 = - r = u6 



a 6 



Hence the formula 



fsin7; .................. (8) 



gives the motion of an infinite mass of liquid produced by an 

 elliptic cylinder of semi-axes a, b, moving parallel to the greater 

 axis with velocity u. 



That the above formulae make the velocity zero at infinity 

 appears from the consideration that, when f is large, S% and S?/ 

 are of the same order as e^Sf or etBrj, so that d^r/dx, d-^rfdy are of 

 the order e~ 2 ^ or 1/r 2 , ultimately, where r denotes the distance of 

 any point from the axis of the cylinder. 



If the motion of the cylinder were parallel to the minor axis 

 the formula would be 



