PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
326 
to be extended to all values of m. It is to be feared there is no simpler mode of 
writing the above result; but the approximate solution of any given case could be 
obtained from it by carefully conducting the approximations. 
Case of an indefinitely thin plane at rest. 
§ 7. We may pass from the above problem to the present by making b indefinitely 
small. If R be the resistance to the current per unit of area 
2Rb = cr 
( 44 ) 
and we have also to enquire what the values of n and n derived— 
(a.) From the former solution will now become. To do so I shall write the equation 
for n in the form 
9 sin 6 — Kb cos 9=0, where 9=nb. 
(I.) When Kb is small the roots of the equation are approximated to, in general, by 
sin 9=0, or 9=jn=/3, say. 
Let 9=/3 fix, then to find x we have 
(l3 fix ) sin x=kI> cos x. 
Expanding sin x and cos x in ascending powers of x, we obtain, after some reduction, 
x in the following series of powers of kIj 
t 7 1 o^o 6+yS 2 R73 
b W Kb + ■ ■ ■ 
moreover 
\ o . o\ 
h=—(r+ri 2 ). 
Hence, in general, 
nb=jn+ l r-Kb -• • • 
J j7r yV 3 3;°7 t 5 
1 
X R / -9 0 , r, 7 . j 2 7T 2 — 1 07 9, 18 + 2/7T 3 R , 
X 0 -jij~ 1T ~fi%Kb- 1- 2 K'lr -o -44 .Kir-. 
Inrb\ J ~tt 6] V* / J 
. (45) 
(2.) But there is a root of the equation for which n b is very small given approxi¬ 
mately by rdbr = k9 ; expanding fully the equation 9 sin 9—Kb cos 0=0 in ascending 
powers of 9, we find 
n i V t =Kb-%K?b*+-* S K?b s + ... q 
X=£(k+!k 2 6+^6 2 + • • •) 
• • («) 
