a 
v and v’ are the velocities of the first and second particles respectively, 
and «is the angle between their directions of motion. We may analyse 
these forces a little differently, and say that the force on the first particle 
consists of — 
1. A force along the line joining the particles equal to 
110 REPORT—1885. 
it l—vv' cose|e? | 
2. A force parallel to the velocity of the second particle and equal 
to 
3. A force parallel to the acceleration of the second particle equal 
to 
ee’ dv! 
@r dt- 
We have, of course, corresponding expressions for the force on the second . 
particle. 
Clausius’ formule differ from those of Gauss, Weber, and Riemann 
in two very important respects. | 
1. They make the forces between two electrified bodies depend on the _ 
absolute velocities and accelerations of the bodies, while the others make 
them depend only on the relative velocities and accelerations. 
2, They do not make the forces between the bodies equal and oppo- 
site, so that the momentum of the system does not remain constant. | 
These results show that if this theory is true, we must take the ether 
surrounding the bodies into account. The first result can then be 
explained by supposing that the velocities which enter into the formule 
are the velocities of the bodies relatively to the ether at a considerable 
distance from the bodies, and the second result by supposing that the 
ether possesses a finite density, and that the momentum lost or gained by 
the bodies is added to or taken from the surrounding ether. 
The case is analogous to the case of two spheres A and B moving in 
an incompressible fluid; in this case the forces on the sphere A depend 
on the velocities and accelerations of B relativeiy to the fluid ata great 
distance from the sphere, and are independent of the velocity and accele- 
ration of A; the forces are not equal and opposite, and the momentum 
lost or gained by the system is added to or taken from the momentum of 
the fluid. At the end of this section we shall see that, if we assume that 
variations in what Maxwell calls the electric displacement produce effects 
analogous to those produced by ordinary conduction currents, we get 
the same forces between moving electrified bodies as are given by Clausius’ 
theory. 
Clausius’ theory is not inconsistent with the principle of the con- 
servation of energy, and we shall see that it does not lead to the same 
difficulty as the theories of Weber and Riemann, viz., that under special 
circumstances a body would behave as if its mass were negative. 
Assuming that in an electric current we have equal quantities of 
positive and negative electricity moving with different velocities, Clausius 
has shown in the paper already cited that his theory gives Ampere’s 
results for the mechanical force between two circuits, and the usual 
