146 THE CORRECTION TO THE LENGTH OF
due to the end thus tell mainly upon the electric forces*, and the pro"b reduced to the electrostatical one of finding the capacity of the term. rod as enclosed in the infinite cylindrical case at potential zero. Bu simplified form of the problem still presents difficulties.
Taking cylindrical coordinates z, r, we identify the axis of symmetr that of 2, supposing also that the origin of z coincides with the flat end interior conducting rod which extends from — co to 0. The enclosing c the other hand extends from -co to -+- oo . -At a distance from the e the negative side the potential V, which is supposed to be unity on. tl: and zero on the case/ has the form
_ log b/r °~ \ogb/a' ................................
and the capacity per unit length is l/(2 log b/d).
On the plane z — Q the value of V from r = 0 to r = a is unity. knew also the value of V from, r = a to r = b, we could treat separate problems arising on the positive and negative sides. On the positiv we could express the solution by means of the functions appropriate • complete cylinder r< 6, and on the negative side by those appropriate annual cylindrical space b > r > a. If we assume an arbitrary value over the part in question of the plane 2 = 0, the criterion of its suit* may be taken to be the equality of the resulting values of dV/dz on tl: sides.
We may begin by supposing that (1) holds good on the negativ throughout ; and we have then to form for the positive side a function shall agree with this at z = 0. The general expression for a function shall vanish when r = b and when £ = + oo , and also satisfy Laj equation, is
A,J0 far) e~
where k1} kZ) &c. are the roots of J"0 (/<$) = 0; and this is to be idei when z = 0 with (1) from a to b and with unity from 0 to a. The coeffi A are to be found in the usual manner by multiplication with J0 (kni integration over the area of the circle r = 'b. To this end we require
J0 (kr) r dr — —j- J0' (kd), ...........................................
i K
t J0(kr)rdr = -^\bJ0'(kb)-aJ0'(ka)}, ............................
J a K
\ log r J, (kr) rdr = -r, [b log b J0' (kb) - a log a/0' (kd)} ~-^-J0 (ka). ..
J a to K
* Compare the analogous acoustical questions in Theory of Sound, §§ 265, 317. p. 199; Scientific Papers, Vol. iv. p. 327.d with 6. The same meth< may be applied to the other formulas (9), (10), (11).