398 Messrs. G. C. Foster and 0. J. Lodge on the Flow of 



Hence for the ratio of the radii vectores from the two poles for 

 the points P, Q, R, S, we have 



fj/ 1 fM n + 1 fjb n + 2 /JU n+S 



r r H r 



respectively ; that is, we have for all these points the common 

 ratio 



fju n ~ m . 



Similarly, we should find for the points P p Qj, and Rj the 

 common ratio 



,,n— m + 1 

 H* i 



and for the points Q ; , R', and S' the common ratio 



,,n— m— 1 

 fj, 



Hence, not only are the equipotential lines for two equal and 

 opposite poles characterized by a constant ratio of the radii vec- 

 tores, as already proved (§ 16), but this ratio changes in a con- 

 stant ratio on passing from any one line of the system to the 

 next, the ratio of change (=/*,) being the same as the ratio of 

 change of radius on passing from line of the system due to a 

 single pole to the next. 



19. The actual potential at any point of the sheet, in terms 

 of the distances of this point from the two poles, follows directly 

 from equation (2) in § 7. Let r be the distance of the given 

 point from the source, and r f its distance from the sink; put V 

 for the potential at the point due to the source alone, and V for 

 that due to the sink alone. Then we have 



and 



V=V' l -^ 8 .log^ 



and since the source and sink are equal, V / 1 =— Y 1 and 

 Q' = — Q ; therefore the resultant potential, or V + V ; , is 



v =£b Ao 4 < 4 > 



This gives the potential =0 at all points of the straight line 

 equidistant from the two poles, positive on the side of this line 

 next the source, and negative on the side next the sink. Also 

 it shows that for equal differences of potential we must have equal 



T 



differences in the value of log -j, which agrees with what was 

 proved above (§ 18). 



