221 
of Edinburgh, Session 1873-74. 
gation of the possible expressions for the forces which satisfy 
Ampere’s data without necessarily satisfying his assumption. Both 
of these authors make the undetermined part of the expression 
depend upon a single arbitrary function. My investigation leads 
to two. The question is one of comparatively little physical 
importance, but I give this investigation for its extreme sim- 
plicity. 
The following is, as nearly as I can recollect, my original pro- 
cess, which has, at least at first sight, nothing in common with 
that of Clerk-Maxwell. 
1. Ampere’s data for closed currents are briefly as follows : — 
1. Reversal of either current reverses the mutual effect. 
II. The effect of a sinuous or zig-zag current is the same as that 
of a straight or continuously curved one, from which it nowhere 
deviates much. 
III. No closed current can set in motion a portion of a circular 
conductor movable about an axis through its centre, and perpen- 
dicular to its plane. 
IY. In similar systems, traversed by equal currents, the forces 
are equal. 
2. First, let us investigate the expression for the force exerted 
by one element on another. 
Let a be the vector joining the elements a 19 a of two circuits ; 
then, by I., II., the action of a L on a' is linear in each of a 15 a', 
and may, therefore, be expressed as 
K, 
where p is a linear and vector function, into each of whose con- 
stituents cq enters linearly. 
The resolved part of this along a is 
S . T5a!tpa , 
and, by III., this must be a complete differential as regards the 
circuit of which a l is an element. Hence, 
(pa' = - (S . af7)if/a' + Va'xoq , 
where if/ and x are linear and vector functions whose constituents 
