DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
4 11 
molecule, the molecule will be split up when the relative velocity of the atom exceeds 
a certain value ; so that in this case the limits of the relative velocity would be 
functions of the coordinates fixing the position of the atoms, and not plus and minus 
infinity, as in Boltzmann’s investigation. We can, moreover, imagine a kind of 
molecule for which we can prove that the theorem itself is not true. We know that 
many dynamical theorems have their most elegant applications to systems of electric 
currents flowing through neighbouring circuits, and that if any dynamical theorem 
is true at all it must be true when interpreted in an electrical sense as well as in the 
mechanical one. This is evident, because we can apply the same method, that of 
Lagrange’s equations, to both the electrical and mechanical problems. Thus 
Boltzmann’s theorem, if it is true at all, must be true when some of the coordinates 
fixing the configuration of the molecule are coordinates which fix the distribution of 
electric currents flowing through circuits attached to the molecule. Let us suppose 
that these coordinates, which we will call x x , x 2 , . . . x n , fix currents flowing through 
perfectly conducting circuits in parallel planes in the molecule, the circuits being so close 
together that the magnetic force due to the currents round the other molecules, or to 
any other external source, may be taken as constant over the circuits. The kinetic 
energy due to the currents x x , x 2 ,. .. circulating through these conductors is of the form 
Ln^i 3 + 2L 13 x ] % -f . . .). 
Let y x , y 2 , be the “ principal ” coordinates, fixing the same configuration as that 
fixed by x x , x 2 , ; then, when the kinetic energy is expressed in terms of these 
coordinates, it is of the form 
+ L 3 y 3 3 + . . .). 
The electrical equations are :— 
J( L ^> 
rate of diminution in the number of lines of force passing through the 
circuit corresponding to y, 
d 
dt 
(L 3 y 3 ) = the same thing for the circuit y 2 , 
Now, since all the circuits are parallel, and so close together that the magnetic force 
may be considered constant over them, the number of lines of force passing through 
y x will be in a constant ratio to the number passing through y 2 . Let this ratio be \ ; 
then 
f(L iyi ) = xJ(L^) > 
or, if y x and y 2 are each initially zero, 
^xUi — ^ 
