342 PROFESSOR C. NIVEN OR THE INDUCTION OF ELECTRIC 
which is satisfied by 
*=!*% .(*•> 
and from the last equation 
d 2 (P + P n) d ~X n 
“P ipdz 
whence 
o- x =w(P+P 0 ).(7) 
By equation (6 X ) this may be written 
s vSp =4 (p+p »). (8) 
This is the characteristic equation of steady currents in a rotating conductor : we 
may show that the first of equations (5) is also satisfied, for this requires that 
o-fAfi cl d(P + P 0 ) a) d~(P + P 0 ) d d /r, , 7 ^ \ 
-pi* =“ T p p —p— + P d? dpPip T+ p «)’ 
or 
— o-0> = aj^(P + P 0 ), 
which corresponds with the former equations. 
All the conditions of the problem will be satisfied by determining P subject to the 
following conditions :— 
(1.) It must satisfy equation (8) within the solid, and vanish when z —— oo . 
(2.) ,, ,, v : P = 0 outside the solid, and vanish when '= + oo. 
dl> 
(3.) P and — must be the same outside and inside the solid when ~ = 0. 
clz 
Let us now suppose that P 0 can be expanded within the solid in a series of the form 
P 0 =S(Ae^+A / e - ' , '" #> )J m (xp)e K ~\ each of these terms verifying the equation v c P 0 =0 as 
they ought to do, and vanishing when z= — oo . We shall show presently how this 
may be done. 
The corresponding part of P due to Ae'"^ is 
where 
P = — (xp) e K: + n 
0 v 2 n = +<o 
•±7T 
dn 
clef) 
( 9 ) 
Putting therefore 
n m {xp)eT- 
(10) 
