ON ELECTRICAL THEORIES. 115 
where A is a constant which depends upon the unit of current, r is the 
distance between the elements, and « the angle between their directions. 
F, E. Neumann showed that this assumption leads to the same law of 
force between two closed circuits as that given by Ampére, and also ex- 
plained by means of it the induction of electric currents. v. Helmholtz ! 
has investigated the most general expression for the energy possessed by 
two elements of current which is consistent with the condition that the 
force between two closed circuits should be the same as that given by 
Ampére’s theory. We shall consider this theory in detail, as it includes 
all theories of this class, and we shall wish to refer to it when we come 
to discuss the relative merits of the various theories. v. Helmholtz 
begins by showing that the most general expression for the energy of two 
elements of circuit consistent with Ampére’s laws for closed circuits is 
Aaa’ 
= {(1 + &) cos « + (1 — &) cos 8 cos 6’} ds ds’, 
A 
2 
where 9 and 6’ are respectively the angles ds and ds’ make with the line 
joining the elements, / is a constant, and the other symbols have the same 
meaning as before. 
Let us call this quantity T ; then we know that T denotes the existence 
of a force dT /dr or 
Af?cd 
t @ 
2 
—} {(1 + &) cos « + (1 — &) cos 4 cos 6'} ds ds! 
along r, and a force — dT/rd0 at right angles to r in the plane of ds and 
#, and in such a direction that it tends to diminish 6; this force equals 
Agata : ‘ do’ 
4 2 (1—k) (sin 4 cos 6’ + cos 0 sin 6’ a) 
and since 
ae 
—-=Ccos 
do UB 
where 7 is the angle between the plane containing r and ds and that 
containing r and ds’, the transverse force 
Azcud! 
= (1—k) {sin 4 cos 6’+cos 4 sin 6’ cos n}. 
=—1 
a 2 
We see that these forces will coincide with those assumed in Korte- 
weg’s theory if the quantities a, b, c, d, which occur in that theory, have 
the following values : 
a= — A? 
fa obey Ae 
22 id =e 
d=4 (1-4) A? 
So that whatever “be the value of /:, these quantities” satisfy the con- 
dition g 
2a+6b+¢—2d= — 3A% 
* Crelle, Ixxii, p. 57; Gesammelte Werke, vol. i. p. 545. 
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