342 
MR. J. J. THOMSON ON DYNAMICAL PRINCIPLES, ETC. 
If we apply Lagrange’s equation to the magnetic coordinate 17 we see that there is 
a magnetising force 
=f( a )y .( 116 ) 
so that when an electric current flows along a twisted wire it magnetises it. This 
phenomenon has also been observed. 
§ 12 . We may complete our survey of the terms in the kinetic energy with this 
example as the term which contains the product of the velocities of the geometrical 
and strain coordinates does not give rise to questions of much interest, and we saw 
that this was the only other product term in the expression for the kinetic energy. 
We have hitherto made the very important restriction that the cases we considered 
were those where there were no resistances, frictional forces, or such things as electric 
resistance, &c, If we give a wide enough meaning to the term material system, we 
ought to be able to deduce such forces from the dynamics of such systems by the use 
of the ordinary dynamical methods. Forces of this kind are assumed to be propor¬ 
tional to the velocities of the corresponding coordinates, and a steady transformation 
from one kind of energy into another, generally heat, is supposed to go on without 
any reverse transformation taking place. It can, however, I think, he proved that 
such forces cannot be deduced from the dynamics of an ordinary system, supposing 
the arrangement of the system to remain continuous. What I believe the equations 
of motion with the frictional forces inserted in the usual way give, is a result which is 
true on the average taken over a time which depends upon the nature of the problem, 
but is not true at any particular instant. Thus, to take the case of electrical resistance 
as an example, we may look upon an electrical current flowing along a wire as the 
limit of what happens when we produce a succession of sparks across an air space by 
means of an electrical machine ; if we suppose the interval between the sparks to 
diminish indefinitely, the discharge will behave like a continuous current, and we may 
suppose a continuous current to be a succession of discharges following each other at 
very short intervals. The ordinary electrical equations w r ith the resistances inserted 
in the usual way will give the mean state of things when the mean is taken over a 
time which includes a good many discharges, but it does not represent the state of 
things at any particular instant. I hope in a future paper to return to the theory of 
this kind of average motion of dynamical systems. 
