122 REPORT—1885. 
The axis of the resultant couple is perpendicular to the element and to 
the vector whose components are U, V, W. 
In another paper! v. Helmholtz discusses the force acting per unit of 
volume on a conductor traversed by electric currents; he shows that, 
according to the potential theory, if w, v, w are the components of current 
through an element dx dy dz, and X, Y, Z the components of the force 
acting on this element of volume per unit of volume, then 
xa + (f-) +0 (S-) 08] 
du dv dw dv de 
E Med =e ‘<a =) + val 
du aw vs TW) | a] 
‘dz ‘y)t at | 
He then discusses the aia of the pa law to sliding contacts, 
that is, contacts such as those made by a wire dipping into mercury; in 
the derivation of the forces from the potential law it is assumed that the 
displacements are continuous, and it might be objected that we have no 
right to apply the law in this case as the motion of the wire and the 
mercury seems at first sight discontinuous. v. Helmholtz, however, 
points out that, as the wire carries the mercury with it as it moves, the 
motion is not really discontinuous and that Neumann’s law is applicable. 
The question of sliding contacts comes prominently forward when we 
compare the various theories; we shall return to it again in this con- 
nection. 
v. Helmholtz also in this paper investigates the electromotive forces 
acting on a conductor in motion; he shows that if the components of the 
velocity of the conductor at any point are a, 3, y, then P, Q, R, the com- 
ponents of the electromotive force, are given by the equation 
Pap (9 — 2) + 7(q ae) ty Oat VB+Wy), 
dy 
with similar equations for Q and R. 
He also investigates the difference between the results of Ampére’s 
and Neumann’s theory for the E.M.F. due to induction. The results are 
complicated ; for practical purposes it is sufficient to notice that when 
there is a mechanical force tending to make the body move in a certain 
direction, there must be an HE. M.F. when the body moves in that 
direction. 
Z=A? [ 
CO. Newmann’s Theory. 
C. Neumann assumes that the electric potential energy is propagated 
with a finite velocity, and that if two electrified bodies are in motion, the 
mutual potential energy is not ee’ /7, where r is the distance between them, 
but ee’/7’, where 7’ is the distance between them at a time ¢ before, 
where ¢ is the time taken by the potential to travel from the one body to 
the other. 
The energy considered in C. Neumann’s theory is a kind of energy 
quite different from any that we have experience of; it is not poten- 
1 Ueber die Theorie der. Elektrodynamik, Crelle, lxxviii. pp. 273-324, 1874; 
Gesammelie Werke, vol. ii. p. 703. 
