102 Proceedings of the Royal Society of Edinburgh. [Sess. 
determine a-^. Thus when '^ = 1T, equation (4) shows that 0^ is 6° 40', 
making 0 ^ = 14° 59', and the value of F at the point of inflexion is 225. 
These numbers are in agreement with the case illustrated by the curve 
15° in fig. 2. 
8. Consider next how the limiting stable deflection 0^ and the corre- 
sponding value of F, namely F^, depend upon the value of i and the 
inclination a of the applied field. In the particular case where ct = 90°, 
equation (2) gives 
(2 + 3 sin^ cos . . . . • (5) 
i 
Hence when the deflecting field acts at right angles to the line of centres, 
by giving the angle of rupture the values shown below, the stated values 
of i are calculated. The table also gives the corresponding calculated 
values of F^, from which may be determined by equation (1). 
Rupture of a Row of Magnets when a = 90°. 
For Qr~ 
30' 
45' 
1° 
2° 
3° 
5° 
10° 
15° 
1-0125 
1-0187 
1-0250 
1 -0506 
1-077 
1-130 
1-274 
1-423 
F — 
2500 
1110 
625 
154 
67 
24 
5-8 
2-6 
Some of these results are plotted in fig. 3. It will be noticed that for small 
values Or is nearly proportional to i—1, that is, to (a — r)/r. Hence 
bears a nearly constant ratio to a — v\ in other words, the line PP' (fig. 1) 
has nearly the same inclination to the line of centres when rupture occurs, 
for all small values of Or- 
When i—1 is made indefinitely small, with the result that Or also 
becomes indefinitely small, the quantity F^, for a = 90°, tends towards the 
2 
limiting value g — ly' show this it is convenient to express the 
deflection of the magnets in terms of the perpendicular distance of P from 
the line of centres (fig. 1). Call that distance x, and write c for JPP'. 
CtCC 
Then x = r sin 0, ON = ^ , and the equation of equilibrium becomes 
Hr sin {a - 0) 
4c^ 
Hence for a = 90° we have, in the limiting case when 0 is indefinitely 
small, 
max 
H = 
4?‘C" 
(Jj f X 
and the criterion for rupture is that ^ ) should be equal to zero, which 
