THE ELECTRIC AHD LUMIHIEEROUS MEDIUM. 
223 
jethereal force (P', Q', R') round a circuit fixed in the (ether lias the same value. 
Again if (F', G', H') be defined so that F' = lujr.dr, we have 
so that 
where {a, /3, y) is magnetic force and V' is the potential of tlie magnetism ; hence 
Ampere’s law follows that the line integral of tlie magnetic force round any circuit 
is equal to 47r times the total current that flows through its aperture. These two 
circuital relations are coextensive with the previous equations involving the vector 
potential, and can thus replace them, when the difference between (P, Q, R) and 
(P j Q j R ) is inessential, that is (i) when the disjilacenient currents are negligible, 
or (ii) when the matter is at rest; tne quantity '4/' then enters as an arbitrary 
function in the integration of the ecjuations. 
The mechanical force acting on the matter, or ponderomotive force, is (X, Y, Z) per 
unit volume, where (§38 infrai) 
fipP. 
The mechanical traction on an interface will be considered later (§ 39). In a 
magnetic medium the magnetic force (a, y) differs from the magnetic flux {a, h, c) 
simply by not including the influence of the local Amperean currents; thus 
a = a — IttA. 
When there is no conductivity, the free charge must move along with the matter, 
so that 
dp . (fiV ,dpq .dpr 
(It dx dy dz ’ 
therefore, from the circuitality of the total current, we must have, identically, 
dt dx \dt dt) ^ dy\dt dt ) ^ ~dd \ dt dt) ' 
Ihe latter is the same as the converg’ence of (fijdt — d/dt) {f, g', h'), which asserts 
(for the case of uniform motion that is contemplated) that mere convection of the 
polarized medium does not produce separation of free electricity. The relation 
between {f, g, h) and (P, Q, R) must be such as to strictly satisfy this equation. 
The quantity occurs in the equations of the field as an undetermined potential 
which is sufficient in order to conserve the condition of bodily circuitality 
dujdx -1- dvidy -f divldz = 0. 
In order to express the conditions that must hold at an interface of transition, we 
notice that by definition F, G, H are continuous everywhere ; but it is only when the 
