38 
SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
The conformal representation of the mapping connecting 2 = x-\-iy and f + is 
given by 2 = c ch ^; and then with tv = 0 + 7 = /B + ia, the relation tv = ch (^—7) 
gives the mapping of <p, \fr, the velocity and stream function of the motion, so that 
( 2 ) (p = ch {tj —a.) cos x/r = sh (>; — «) sin 
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 
for < a on the Riemaim 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 >/ = a, and then emerging in another 
sti'eam off to infinity. 
The Weir chart would be ideal to employ in plotting some ot the curves described 
by Legendre (F.E., I., p. 411), orbits invented by Euler, 1760, under two centres 
of gravitation at S, S', as the variables employed by Euler and Legendre are 
tan - 2 ^, th and these are separated in the equations of motion. 
