1886.] on an Element of a Magnet carrying a Current. 41 
where there is magnetisation. Now as we have just shown inci- 
dentally 
V.CH,= V.68 — V. 225. 
So Maxwell tacitly assumes that there is a force V.@27% acting 
from the element considered as a magnet, on the element con- 
sidered as a conductor. 
It is quite permissible to do this, in order to simplify the 
mathematical expressions, provided that, in reckoning up the 
forces acting on the element in virtue of its magnetism, we do 
not forget to include a force — V.@2 due to the current tra- 
versing the element. The addition of this force changes the 
x-component from 
Wh d (a, +a, +4,) 4B d (a, +a, + ,) i a? (ay + a, +45) 
dx d dz 
ip 9 May as 
ae st dx dx AEB 
as we have just shown. 
If we reverse this imaginary force, we find the force on the 
element considered as a conductor is V.@, and the z-component 
of the force on it considered as a magnet is 
da dx dx 
Heda to aa die 
The last is the same as we should obtain from first principles, 
if it were given that the magnetised element were placed in an 
undisturbed field S. 3 
This pair of expressions is just as correct as the pair given by 
Maxwell, and leads of course to the same value of the total force 
on the element. 
It might be contended that it would be more logical to suppose 
the imaginary force zero, but it is easy to show thatthe value of 
the imaginary force depends on the form of the element. The 
total force on an element, it 1s obvious, must depend merely on 
its volume, but the same is not true of the parts into which we 
arbitrarily divide the total force. In the case of a spherical element 
V.CS= V.6H,+ V.C 8x, 
so the imaginary force in Maxwell’s expressions has the value 
V.C873. 
To obtain Maxwell’s expressions without any imaginary force 
coming in, we must use as an element a thin disc at right angles 
to the magnetisation, and to obtain the expressions suggested 
above, we must use a thin cylinder parallel to the magnetisation. 
With these elements it is easy to find the force on the element 
