782 
Proceedings of the Royal Society 
t- = x + e - = _ A +1+ 2e -'fe = - he + 1 + «. 
7i 2 
\ } or \ x = X 3 , besides A 2 = 0; 
A. 
whence 
A, A, 
e — — e —* 
Aj „c: A, 
+ e - J e 
A, 
©' 
+ A _ 2 
Aj 
.A 
i.e., = 1, or - 2 
A i 
0 
or 0 . 
Thus p 2 = 7V 1 , or n 2 (l + 3e), or w 2 (l + e). There is no farther 
difficulty in applying the method to magnets of different masses or 
magnetic strengths; but it is interesting to observe that, by pro¬ 
perly adjusting the gaps in terms of the masses and magnetisation 
of the bars, any set of magnets whatever can be brought to behave 
(for small oscillations) as if they were in all respects equal to each 
other and arranged at equal distances. 
"When there is an infinite series of magnets arranged in this way 
the equation above may be written 
[(*) 
-tt 1 + n ‘ 
p - iy 
a 3 
D 
] 
X r 
= 0, 
where 
Vx r = x r + 1 , 
of which the general integral is easily found. 
When the number of magnets (m) is finite, and they are arranged 
in a closed curve, we have the conditional equation 
(D m - V)x r = 0. 
In this case the general solution may be elegantly expressed in 
terms of the m th roots of unity. It leads to some curious proper¬ 
ties of determinants, whose development will form an excellent 
exercise for the student. Thus, writing in succession 1,2, ...., m 
for r ; and putting 
cr 
d 
2 
