106 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and 
H 
m 
4/^2 
4 cos2 ^ sm(cf3 + 0) 
(2^ - cos 6 - cos 0')^ ' sin (a + 0) 
] 
( 9 ) 
Here again the quantity in square brackets is a numerical factor, which we 
shall write F'. The condition for rupture is that = 0, and this, together 
do 
with equations (7), (8) and (9), makes F/ determinate. When d and 0' at 
rupture are very small, as will be the case when a is only very little greater 
than T, (p at rupture becomes equal to a and the value of F/ is 
cos^ a 
(^-l) 
2’ 
1 
, as it should do to agree with the 
which when a=180° becomes 
(i— ly 
earlier consideration of that simple case. From the maximum at a =180° 
the curve of F/ falls away to either side proportionally to cos^ a when 
7—1 is indefinitely small, and at a rate which is nearly proportional to 
cos2 a for such values of i as we are concerned with, until it meets the 
curve of F^ in a point of infiexion (as at C in fig. 4), which is the point at 
which the mode of rupture changes from that of fig. 1 to that of fig. 5. 
At that point both modes are equally probable. When 7 — 1 is very small 
this change-point is at a = 145°, as will be seen from the fact that F^. then 
2 1 
and F/ to the limit- 
approximates to the limiting value 
3x/3(7-l)2 sin a 
To make these equal requires a to be 180°±35 
With 
. cos^ a 
ing: value ^ 
- (^-lp 
larger values of 7 the point of inflexion is shifted a little nearer to the 
180° position. 
11. All these conclusions are applicable with a very simple change to 
a solitary pair of magnets instead of an indefinitely extended row. In a 
row the restoring force acts on both poles of each magnet ; in a solitary 
pair the restoring force acts on one pole only, while the deflecting force 
still acts on both poles. Hence to maintain a given deflection the value 
of H for the pair is only half its value for the row. This applies in both 
modes of rupture. Hence for a pair of magnets the field which will 
produce rupture is H,. = g— ^[FJ from the smallest value of a, namely up 
to the value at which the mode of rupture changes ; and from that point 
111 
up to and through the maximum at a = 180° it is H^ = g~ 2 [E/]. The 
deflection at rupture (for any value of a and of 7) is the same for the pair 
as for the row. 
12. In order to confirm and illustrate these conclusions the magnetic 
