348 
PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
Eliminating C and D we find 
B 1 S« +1 +BoT ii+1 — 0 
b 1 s / *« 1 +b 2 t , *_ 1 = — 
2n+l 
0 ’ 
whence we find for C, which is what we chiefly want, 
C=A -1 
Li + 1 S„ +1 T « —T ii+1 Sh ] _ , S„+iT )i+I T, t+1 S , t+1 (ot\\ 
•Q TV rn QC/ f rp/ _-p Q/ 
^/e+l- 1 - n —i n—\ J n-\ n—\ 
/3 
(b.) When the inducing’ magnetism lies partly within the shell (not in its substance) 
w T e may take for P 0 a series of the form 
AV'^P,/ 
b\" +1 
the other expressions for P will retain their forms except that we must take the first 
term —A'( instead of — A( -) . The calculation of C is similar to that given above; 
and, wdien the algebraic work is performed, we obtain finally 
c= v , _ 1 _2^+!/bV +1 SbT'^-TbSk.! 
Xa \a/ S„ +I r—T, (+ 1 Sb_] 
(27) 
There are two particular cases of these formulae which possess a special interest. 
(I.) In the case of a solid sphere a=0, S« +1 (\a) = 0. 
The inducing magnetism can only belong to type (a) and may be represented by 
P 0 =SAe' ,w TV L-j : the external effect of the currents is then given, according to 
(26), by 
P — V A ^»+l( ^ h ) 
by+i 
(28) 
and it is easy to deduce from this expression the value of P when P 0 is given by terms 
which are real in <f>. 
(II.) In the case of an infinitely thin shell, we must put b = a+c. 
Tins case is interesting as it constitutes an extension of Maxw t ell’s result for a 
plane plate. A glance at the preceding formulae will show that we have here to 
expand S„ (Xa+Xc), T„(Xa + Xc), where v—n — I, n or n + J, in ascending powers of c. 
Some caution is requisite, however, in doing so on account of the value of X: for 
