322 
MR, J. J. THOMSON ON SOME APPLICATIONS OF 
so that the existence of T implies a force parallel to x equal to dT/dx, but if T is tire 
energy expressed in terms of the momenta instead of the velocities, 
dT'__tfT 
dx dx’ 
so that the force parallel to x —— dT'/dx. As in our case the energy is expressed in 
terms of the momenta, and not of the velocities, we see that the force parallel to x 
(32) 
(33) 
the usual expression, though (32) is preferable.'"'] 
{ zz } cannot be a function of the electrical coordinates y, because if it were electro¬ 
motive forces due to magnetised bodies would exist which would not be reversed 
when the magnetism of all the bodies in the field was reversed. 
{zz} cannot be a function of the quantity £ which helps to fix the intensity of 
magnetisation, for if it were the kinetic energy would no longer be a quadratic 
function of £ but would involve higher powers. It may, however, be a function of y, 
and will be so if the coefficient of magnetic induction depends upon the intensity of 
magnetisation. As experiments show that this is the case we conclude that {zz} is a 
function of y. 
A large number of experiments have been made in order to find the value of the 
coefficient of magnetic induction for all values of the intensity of magnetisation; all 
these experiments agree in showing that the coefficient diminishes as the intensity of 
magnetisation increases, so much so indeed that there is a maximum value of the 
intensity of magnetisation which cannot be exceeded however intense the magnetising 
force may be. As the experiments are not sufficiently consistent to lead to the 
determination of the law connecting the intensity of magnetisation with the coefficient 
of magnetic induction, all that we can do is to take some empirical law which agrees 
with one set of experiments and deduce its consequences. It is probable that the 
results will in their main features agree with those which could be deduced from the 
true law. For our purpose we shall take the empirical law proposed by Stefan 
(Wiedemann’s ‘ Lehre von der Elektricitat,’ vol. 3, p. 432), which is expressed by the 
equation 
= I 
dX ? 
=1 
dx 
or since I = KX 
=iA KX2 * 
* This paragraph lias been re-written since the paper was sent in to the Royal Society. 
