328] Conjugate Functions 271 



At a point on the equipotential F=0, the surface-density is 



4 



-7T 



At P, 7 =-oo, so that " = 4p a we approach Q, a increases and finally 



becomes infinite at Q, while after passing Q and moving along QR, the upper 

 side of the plate, <r decreases, and ultimately vanishes to the order of e~ u . 



The total charge within any range U lf U 2 is, by equation (233), 



It therefore appears that the total charge on the upper part of the plate QR 

 is infinite. 



Let us, however, consider the charges on the two sides of a strip of the 

 plate of width I from Q, i.e. the strip between x = h/jr and x = I + hjir. The 

 two values of U corresponding to the points in the upper and lower faces 

 at which this strip terminates, are from equation (244) the two real roots of 



Of these roots we know that one, say U lt is negative and the other (U 2 ) is 

 positive. If I is large, we find that the negative root U^ is, to a first approxi- 

 mation, equal to 



and this is its actual value when I is very large. Thus the charge on the 

 lower plate within a large distance I of the edge is 



and therefore the disturbance in the distribution of electricity as we approach 

 Q results in an increase on the charge of the lower plate equal to what would 

 be the charge on a strip of width h/jr in the undisturbed state. 



If I is large the positive root of equation (245) is 



so that the total charge on a strip of width I of the upper plate approximates, 

 when I is large, to 



Thus although the charge on the upper plate is infinite, it vanishes in 

 comparison with that on the lower plate. 



