An example of the equipotential curves in the ajy-plane is shown in Figure 220, where 

 Cj = c, C2 = - c and the heavy curve is C or the trace of the dividing surface of revolution. 



Further mathematical details will be given only for the case in which the line of 

 dipoles extends to infinity in one direction. 



Half-Infinite Solid of Revolution 



Let the dipoles extend from a? = to a; -♦ »=. Then = 6, where is the polar angle 

 at the origin, cos d„ = - 1 and 



V (D + — (1 + COS 



[133c] 



1/2 



Here cos 6 = x/r, r = {To + x ) , wr = r sin 6. Hence, 



9 = - -— = cos w, 



dx .3 



[133d] 



do) 



-V + (1 + cos d + sin d cos 



—2 



cos CO, [133e] 



?.., = - 



"ST do 



V + (1 + cos 



~2 



[133f] 



?r = 



d4> I V 1 + cos 6 \ 



— = -V sin + cos (o, 



dr \ ,2 sin d / 



[133g] 



1 dcf) I V \ + cos 6 \ 



— = I -^ cos + — cos o), 



r dd \ ,2 gjj^2 Q / 



[133h] 



1 



?.„ = - 



rsin^ d(o 



v 1 + cos 6 \ 

 V + — jsin oj. 



r^ sin^ (9 / 



[133i] 



The speed q may be found from q^ = q^ + q^+ q^ - q^ + 



2 2 

 9 + ?., 



As a; -» ix. r -» <x, -+ 0; hence (? -♦ 0, and, at &r= J 2 i^/F , (7^= while o, = 2 F sin co. 

 The streamlines are thus tangent at a? = + ~ to a cylinder of diameter 2\/2 v./V . The solid of 

 revolution must, therefore, be asymptotic to this cylinder. 



Some of the equipotential curves and streamlines in the a-y-plane are shown in Figure 

 221; the heavy curve is the outline of the solid. The equipotential curves are extended in- 

 ward toward the line of dipoles. (In this figure V is denoted by V.) 



334 



