_ 1886.] | on an Element of a Magnet carrying a Current. 39 
currents. Then the element is suspended in a field of force §,, 
so the forces on it are expressed by (3) and (4), with a,8,y, written 
for aBry. 
Now we know that , within the sphere is uniform and 
parallel to the magnetisation, so both the force (3) and the couple 
(4) vanish when a8, are written for aBy. Hence (3) and (4), 
as they stand, correctly express the forces on the element. More- 
over it is obvious that the actual forces must be proportional to 
the volume and independent of the form of the element. 
Now let us consider the general case when a magnetised element 
is traversed by a current. In this case let the element be a long 
thin circular cylinder whose length is parallel to the current. By 
similar arguments to those used before, we may consider the cur- 
rent and the magnetisation to be uniform. We shall avail ourselves 
of the following results, which are well known or may be easily 
proved. In an infinite circular cylinder uniformly and transversely 
wig Besa, 
In an infinite circular cylinder, uniformly magnetised longitudi- 
nally, § has no transverse component. In an infinitely long 
circular cylinder, traversed longitudinally by an uniform current, 
| 7 
Sor &e. | 
dy dz | - 
da _dB _dy_, | ERS Bite eee 2 
eo dir Ge | 
Tn our cylindrical element now let § be divided into four parts, 
_ §, due to the current in the element, 
§, due to the transverse magnetisation in the element, 
H, due to the longitudinal magnetisation in the element, 
§, due to the external currents and magnetisation. 
It is proved by Maxwell (Art. 490) that the force on the 
element from outside due to the presence of its current has the 
components 
A= Jig) = 103, SiC: penta rabs ib siclaats sar ae (2); 
the force due to the presence of its magnetisation has the com- 
ponents 
d (a,+ a,+ a,) d (at Oy t 4) A(a,+ a+) 9 
da 
AXA=A +.B “ap Sb es aa Pines H@10), 
by the result last obtained. 
