192 1-22. J On Models of Ferromagnetic Induction. 
99 
depends on the inclination a of the applied field. For small values of a 
there is no instability. 
4 Consider the equilibrium of any one magnet in the row. The 
deflecting moment due to H (fig. 1) is 2Hm. OM. The restoring moment 
due to adjacent poles is ~ ^pp^y 2 • Since a — r is small compared with r, 
only adjacent poles need be taken into account. The equation of 
equilibrium is accordingly 
m . ON 
H . OM = 
(pp')2 • 
But OM = OP sin OPM=r sin (a — d), (PPd^^4AP^ = 4(a^ + r^ — 2ar cos d), 
OA.OPsinAOP arsind 
and ON = OA sin OAP = 
AP 
equation of equilibrium becomes 
H sill {a - $) = 
or, writing i for afr, 
{o? + — 2ar cos d)^’ 
77ia sin 0 
Hence the 
m 
4(a^ +?’2 - 2ar cos /9)t 
i sin 0 1 
4- 1 - 2^' cos 6Y ' sin (a - 9)_ 
( 1 ) 
For brevity we shall write F for the quantity in square brackets. It 
is a numerical factor, a function of i and a and of the deflection d, 
increasing from zero with 6. When values of i and a are assigned, F 
reaches a maximum, say F^, for a particular value of d, say d^. At that 
do 
deflection is infinite, and consequently Or is the angle of rupture and 
determines the field which will produce instability. 
5. For example, if i is 1*1 and a is 20° the values of F calculated 
from the above expression for various values of d are as follows : — 
Values of F for ^ = 1-1 and a = 20°. 
d. 
F. 
e. 
F. 
f 
56-1 
10 
121-6 
2 
102-9 
12 
117-4 
3 
132-7 
15 
131-9 
4 
145 9 
17 
177-7 
5 
148-8 
18 
241-3 
6 
145-2 
19 
438-5 
8 
132-3 
20 
00 
These are also shown in the curve, fig. 2, from which it will be seen that 
as H is increased from zero the deflection is at first nearly proportional 
