38 



SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 



The conformal representation of the mapping connecting z = x + iy and f = ^ + ig is 

 given by z = c ch f ; and then with w = <j> + iijs, y = fi+ia, the relation w = ch (f y) 

 gives the mapping of <f>, \jr, the velocity and stream function of the motion, so that 



(2) 



= ch (y a.) cos ( /3), 



= sh (>; ) sin ( /3). 



Fig. 3 gives the ordinary stream motion of the distortion of a uniform current by 

 an elliptic post, and should figure as the typical diagram in a treatise on Hydro- 

 dynamics. But fig. 4 is curious in showing the analytical prolongation of the functions 



Fig. 2. 



for , < on the Riemann sheet, with the cut along the line SS' joining the foci. It 

 shows the middle stream coming to a waterfall across SS' and circulating in a 

 whirlpool chamber in the interior of the ellipse n = a, and then emerging in another 

 stream off to infinity. 



The Weir chart would be ideal to employ in plotting some of the curves described 

 by LEGENDRE (F.E., I, p. 411), orbits invented by EULER, 1760, under two centres 

 f gravitation at S, S', as the variables employed by EULER and LEGENDRE are 

 tan ^, th y,, and these are separated in the equations of motion. 



