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PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
If we can express O in the form 
fl = (4+B|+c£)P. 
where P satisfies the equation v 2 P=0, the equations are satisfied by 
and, in particular, if 
(14) 
(15) 
(16) 
These expressions (15) may be easily verified by actual differentiation, taking account 
of v 2 P=0, and may be looked upon as a generalisation of the equations given by 
Maxwell (‘Electricity,’ Art. 405). They give additional interest to his expression for 
a solid negative harmonic 
<+K + < 
■ ( A "<fe +B 4 +G *s)v ; 
and in forming F, G, H any one of the n factors may be chosen to furnish A, B, C ; 
the results obtained by taking two different factors differing by quantities of the form 
dy dy dy _ . 
7 > yw as I have verified. When fl is a solid harmonic of positive degree, it is 
CtoC CL If CCZ 
more convenient to use the simplified form (16), and in the case of a tesseral solid 
harmonic, the results of differentiation give rise to a series of very interesting 
theorems, which, however, do not interest us at present. 
It will be observed that the expressions given for F, G, IT satisfy the equation of no 
convergence 
dFd_G 
dx dy 
= 0. 
( b .) When we use semipolar coordinates p, (p, z, given by 
x=p cos (p , y—p sin (p 
(17) 
