136 REPORT—1885. 
thus Helmholtz’s equation seems to be inconsistent with the principle 
that the change in the quantity of free electricity is caused by conduction 
currents. In the case above considered, Maxwell’s equations lead to no 
difficulty ; it does not foilow, however, that Maxwell’s assumption that 
the total current behaves like the flow of an incompressible fluid is 
absolutely necessary. We shall consider later on the differences which the 
abandonment of this assumption will make in the theory. 
We shall now go on to consider Helmholtz’s equations and compare 
them with the corresponding ones in Maxwell’s theory. 
The quantities U, V, W are given by equation of the form 
—1 (py) ib ay 
U=4 (1-4 4 + |[[eaa dt, 
where & is the constant which we mentioned before as occurring in 
Helmholtz’s theory, and 
= Lae D 5, 
pa b[fep avi 
where ¢ is the electrostatic potential ; it follows from these equations 
that 
dz dy dz dt 
The corresponding equation in Maxwell’s theory is 
a , av, aw_y 
de dy da ”’ 
so that these equations coincide if s=0. We can see from the value of x 
given on page 116 that, on Helmholtz’s theory, this quantity would also 
vanish, whatever be the value of k, if the total current behaved like the 
flow of an incompressible fluid. 
If a, 6, y are the components of the magnetic force, then on Helm- 
holtz’s theory 
aya {Oe — daw \, 
= i 
dy dz dtd 
da_ dy _ { ap \ 
dz rs dt dy aitigo 
dp da ae \ 
PHA —4 
dx dy didz J’ 
where A is a quantity depending on the unit of current adopted, and is 
such that the force between two parallel elements of currents at right 
angles to the line joining them is 
2 
i Gaede. 
where r is the distance between the elements, 77 the current through them, 
and ds ds’ their lengths ; the corresponding equations on Maxwell’s theory 
are 
dy _ dB _ 
mee oe 
with similar equations for v and w. 
