Length of Terminated Rods in Electrical Problems. 3 



be identified when z = () with (1) from a to b and with unity 

 from to a. The coefficients A are to be found in the 

 usual manner by multiplication with J (&«?') and integration 

 over the area of the circle r — b. To this end we require 



( a j (kr)rdr=-p '(ka), (3) 



. o *-' 



^ b J {kr)rdr=-^(bJ '(kb)-aJ '(ka)\, . (4) 



flog r J O) r dr= - ~ (b log b J '(kb) 



— a log a J '(l:a) j — p J (^) • • ( 5 ) 

 Thus altogether 



^^^\'(kr)rdr=WAJ ^B). . (6) 



For J ' 2 we ma} r write J x 2 ; so that if in (2) we take 



2J {ka) 



Pb 2 ~J \\kb) log b/a' K J 



we shall have a function which satisfies the necessary 

 conditions, and at z = assumes the value 1 from to a and 

 that expressed in (1) from a to b. But the values of dV/dz 

 are not the same on the two sides. 



If we call the value, so determined on the positive as well 

 as upon the negative side, V , we may denote the true value 

 of Vby V + V. The conditions for V will then be the 

 satisfaction of Laplace's equation throughout the dielectric 

 (except at ^ = 0), that on the negative side it make V' = 

 both when r = a and when r=b, and vanish at z=— oo , and 

 on the positive side V' = when r = b and when c=+oo, 

 and that when :=0 V assume the same value on the two 

 sides between a and b and on the positive side the value zero 

 from to a. A further condition for the exact solution is 

 that dV/dz, or dV /dz-\-dV/dz, shall be the same on the two 

 sides from r = a to r = b when ,c = 0. 



Now whatever may be in other respects the character of 

 V on the negative side, it can be expressed by the series 



V^H^M^ + Ha^r) e h r + ..., . (8) 



where </>(/ii?*) &c. are the normal functions appropriate to 

 the symmetrical vibrations of an annular membrane of radii 



B2 



