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 



t = -^sm0 ........................ (3), 



which agrees with Art. 68. 



3. Let us introduce the elliptic co-ordinates f, 77, connected 

 with x, y by the relation 



x + iy = c cosh (% + irj) ..................... (4), 



or x = c cosh f cos 77, j 



y = c sinh f sin 77 J 



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

 77 from to 2?r. If we now put 



+ ty = Cer<M-*i> ........................ (6), 



where (7 is some real constant, we have 



i|r = -Ck-sm77 ........................ (7), 



so that (1) becomes 



Ce~% sin 97 = uc sinh ( sin 77 + 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 

 a = ccosh , & = csinhj- , 



u6c , fa 4- 6\* 



so that C = - r = UO - r 1 . 



a b \a-bJ 



Hence the formula 



sin7; .................. (8) 



a 



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, 8% and 8y 

 are of the same order as e^Sf or 0*877, so ^ nat d-^jdx, d^rjdy 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 



