100 REPORT—1885. 
principle that the magnetic action of an electric current was the same as 
that due to a magnetic shell bounded by the circuit and magnetised to 
the proper intensity. In this way Ampere gave a complete theory of the 
action of currents upon currents and upon magnets—in fact, a complete 
theory of all the effects produced by a current which were known when 
his paper was published. 
It is difficult to overrate the service which Ampére’s theory has 
rendered to the science of electrodynamics. Perhaps the best evidence 
of its value for practical purposes is the extreme difficulty of finding any 
experiment which proves that it is insufficient. In spite of this, how- 
ever, as a dynamical theory it is very unsatisfactory. If, as we are led 
to do by Ampére, we attach physical importance to elements of current, 
and regard them as something more than mathematical helps for calcu- 
lating the force between two closed circuits, then we are driven to ask, 
not only what is the law of force between the elements, but what is the 
energy possessed by a system consisting of two such elements. If we do 
this, and find this energy by calculating the amount of work required to 
pull the elements an infinite distance apart, we arrive at the conclusion 
that the energy must depend upon the angles which the elements make 
with each other and with the line joining them; but if this is so, then 
the force between the elements cannot be along the line joining them, 
and there must in addition to this force be couples acting on the elements. 
For these reasons we see that Ampére’s theory cannot give the complete 
action between two elements of current. What it does—and this for 
practical purposes is an advantage and not a disadvantage—is to give 
in most cases, instead of the complete action between two elements, that 
part of it which really affects the case under consideration. 
Before discussing cases, however, in which the terms which Ampére 
neglects might be expected to produce measurable effects, we shall, in 
order to compare the various theories more easily, proceed to consider 
other theories of the same class. 
Grassmann’s Theory.! 
The method by which Grassmann obtains his theory is very remark- 
able. He objects to Ampére’s formula for the force between two elements 
of current, because it makes the force between two parallel elements 
change from an attraction to a repulsion when the angle which the ele- 
ments make with the line joining them passes through the value cos! 2/3, 
and the object of his investigation is to get a law of force free from this 
peculiarity, and which, while giving the same result as Ampére’s for closed 
vircuits, shall yet be as simple as possible. He begins by regarding any 
circuit as built up of ‘ Winkelstréme,’ 7.e., currents flowing along the two 
infinite lines which form any angle. He points out that a circuit of any 
shape can be built up of such currents; the circuit abc, for example, 
may be regarded as built up of the ‘ Winkelstréme’ eaf, fbg, and gce. 
Grassmann proceeds to calculate by Ampére’s formula the action of 
a ‘Winkelstrom’ upon an element of current (a). Since the ‘ Winkel- 
strom’ will have no action upon an element of current perpendicular to 
its plane, we see that it is only necessary to calculate its action upon the 
component (a’) of ain its own plane. Grassmann does this by calcu- 
lating the effect due to each arm of the ‘ Winkelstrom’ separately. He 
1 Poge. Ann. 64, p 1, 1845; Crelle, 83, p. 57 
