DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
327 
1/B contains powers of e higher than the first, the terms which involve e may be 
written in the form 
ae+/?e 2 + . . . 
Thus, if we neglect the terms containing higher powers of e than the second, d(l/B)de, 
which equals a+2 fie, will change sign when e=—a/2 fi. Equation (46) shows that 
a soft iron bar will expand or contract on magnetisation according as the strain e in it 
makes the expression a-[-2fie negative or positive; this changes sign when e passes 
through the critical value —a/2 fi, so that a soft iron bar when strained beyond a 
certain limit will contract instead of expanding when it is magnetised. Some ex peri' 
ments which Joule made on the effect of magnetising soft iron bars which were under 
great tension confirm this result. Joule found (Sturgeon’s “ Annals of Electricity, 
1842,” Phil. Mag., 1847) that when soft iron wires were stretched beyond a certain 
limit they became shorter instead of longer when they were magnetised; since the 
limiting strain is an extension, the limiting value of e, —a/2/3 must be positive, so 
that a and (3 must be of opposite signs; and as a is negative when the magnetisation 
is intense, /3 must in this case be positive, and if (3 be positive the coefficients of 
elasticity will be increased as we can see from equation (41), so that we conclude that 
the elasticity of a soft iron bar will be diminished by strong magnetisation, i.e., the 
same force will not stretch it so much. 
A great many relations exist between torsion and magnetism, but many of these 
are relations between permanent magnetism and permanent set, both of which are 
outside the scope of this paper. We shall confine ourselves to the relations existing 
between temporary magnetisation and twists which are not large enough to give the 
body any permanent set. Since the magnetisation of a soft iron bar is altered by 
twisting it, 1/B (using the same notation as before) must be a function of the torsion 
coordinates a, b, c: let us suppose as before that the length of the bar is parallel to 
the axis of x, then the coordinate a will fix the rotation of the bar round this axis. 
The couple tending to twist the bar is by the modified Lagrangian equation for a, 
equal to dT'/da, and this equals 
_i! I 1 !) 
z iki\B ' 
if oj be the angle through which this couple twists the bar and n the coefficient of 
rigidity, 
1 d_ /F 
2 n da\ B 
(5G) 
When a twisted bar is magnetised it untwists to a certain extent (Wiedemann’s 
£ Lehre von der Elektricitat/ vol. 3, p. 692), but if an untwisted bar be magnetised it 
does not twist at all; this shows that if 1/B be expanded in ascending powers of 
a, the first power must be absent, for if it were present an untwisted rod would 
become twisted when it was magnetised. By a similar investigation to that by 
2 u 2 
