and Electric Waves upon Small Obstacles. 41 



Thus at a distance r great in comparison with X we have 



p- ikr , \ i 



^ = 7 + log {&ka)\2ifa-) ' ' ' ' ^ 



When the section of the obstacle is other than circular, a 

 less direct process must be followed. Let us consider a circle 

 of radius p concentric with the obstacle, where p is large in 

 comparison with the dimensions of the obstacle but small in 

 comparison with \. Within this circle the flow may be 

 identified with that of an incompressible fluid. On the circle 

 we have 



<f> + f = l + So\y + log(Ukp)l, . . (77) 



2>n-d(<j> + f)/dr = 2TrS 0} .... (78) 



of which the latter expresses the flow of fluid across the 

 circumference. This flow in the region between the circle 

 and the obstacle corresponds to the potential-difference (77). 

 Thus, if R denote the electrical resistance between the two 

 surfaces (reckoned of course for unit length parallel to z), 



S J7 + log(i/fy)-27rR} = l, . . . (79) 



and i|r=S D (/c?'), as usual. 



The value of S in (79) is of course independent of the 

 actual value of p, so long as it is large. If the obstacle be 

 circular, 



2irR = log(p/a). 



The problem of determining R for an elliptic section 

 (a, b) can, as is well known, be solved by the method of 

 conjugate functions. If we take 



x =c cosh £ cos 7], y = c sinh £ sin rj, . . . (80) 



the confocal ellipses 



+ ^= c * (81) 



cosh 2 f sinh 2 £ 



are the equipotential curves. One of these, for which £ is 

 large, can be identified with the circle of radius p, the relation 

 between p and £ being 



£ = log (2p/c). 



An inner one, for which £=£ 5 is to be identified with the 

 ellipse (a, b) } so that 



a = c cosh £ , b = c sinh £ , 



