ON ELECTRICAL THEORIES. 99 
two small magnets is along the line joining their centres, and in this case 
the assumption is untrue. It is in the nature of the assumption V. that 
Ampeére’s theory differs from others of this class. The second part of 
I. depends upon V. It is not true unless we assume that the force 
between two elements is along the line joining them. 
Ampére deduces the force between two elements of current from these 
principles in the following way :—Suppose we have two elements of current 
of lengths ds,, ds. traversed by currents of strengths 7, j respectively. 
Let us take the line joining the centres of these currents as the axis of a ; 
let the plane of ds, and x be taken as the plane of wy; let 6), 9. be the 
angles which ds), ds) respectively make with the axis of z, » the angle 
which the plane through ds, and 7 makes with the plane of ay. 
By Ampére’s second proposition the action of ds, on ds, will be the 
sum of the action of 
ds, cos @, or a, along 
ap sin 6, or , along 
on 
J ds cos 0, or a, along 
ds, sin 4 cos n or [3g along y 
ae sin 6, sin 7 or Yq along z. 
Now by proposition I. «, cannot exert a force on either (3, or yo, 
because it is in planes which bisect 3, and y, at right angles, so that the 
only component on which a, can exert aforceis ay. Let the force between 
these components be 
where 7 is the distance between the centres of the elementary currents. 
__ In the same way we can show that the only component on which /, 
can exert any force is 3,. Let the force between these two elements be 
iS PPro. 
T 
Thus the force between the two elements ds,, ds, is 
= {aaa + BB Ao}, 
2 
or, substituting for ajay, (2,3, their values: 
J: {a cos 0, cos 0. + b sin 0, sin 0, cos n} 77 ds, dsg. 
The proposition III., that the action of a closed circuit on an element of 
current is always at right angles to the element, leads on integration to 
the condition 
2a +b=0, 
so that the force between the two elements equals 
os {cos 6, cos 6,—2 sin 0, sin 0, cos 7} ij ds, dsy. 
From this we are able to find the force between any two circuits or parts 
of circuits. To find the force on a magnetic system, Ampére used his 
H2 
