224 
Proceedings of the Royal Society 
In the ordinary notation this expresses a force whose components 
are proportional to 
(1.) Along a 
(Note that, in this expression, r is the distance between the ele- 
ments.) 
(2.) Parallel to a' 
ds x 
(3.) Parallel to o, - d f,. 
If we assume f = F = - Q, we obtain the result given by Clerk- 
Maxwell ( Electricity and Magnetism , § 525), which differs from 
the above only because he assumes that the force exerted by one 
element on another when the first is parallel and the second per- 
pendicular to the line joining them is equal to that exerted when 
the first is perpendicular and the second parallel to that line. 
7. What precedes is, of course, only a particular case of the 
following interesting problem : — 
Required the most general expression for the mutual action of two 
rectilinear elements , each of which has dipolar symmetry in the direc- 
tion of its length , and which may be resolved and compounded accord- 
ing to the usual kinematical law. 
The data involved in this statement are equivalent to I. and II. 
of Ampere’s data above quoted. Hence, keeping the same nota- 
tion as in § 2 above, the force exerted by cq on a must be ex- 
pressible as 
<pd 
where <p is a linear and vector function, whose constituents are 
linear and homogeneous in cq ; and, besides, involve only a. 
By interchanging cq and a, and changing the sign of a, we get 
the force exerted by a on cq. If in this we again interchange cq 
and a , and change the sign of the whole, we must obviously repro- 
duce (pa'. Hence we must have pa' changing its sign with a, or 
p>a — PaScqa/ + QaSacqSaa + RcqSaa / + Ra'Sacq 
where P, Q, R, R are functions of Ta only. 
