Conservation and Innovation : Helmholtz's Struggle with Energy ...

and unaware of Green's results190, introduced the concept of electric potential. He
defined the quantity
⎛ eieii − ⎝ r ⎠ ⎞
corresponding to the sum of the tension forces consumed, and also of course to
the living forces acquired in the motion of the two charges from an infinite
distance to the distance r, as the potential of the two electrical elements for the
distance r 191.
The principle of conservation can thus be expressed in a new way: "the increase
of vis viva in whichever movement must be considered equal to the difference of
the potential at the end of the trajectory respect to the potential at the
beginning" 192 (the sign of the potential is opposite to that of modern convention).
The potential as defined by Helmholtz is equivalent but for the sign to what later
was to be called potential energy. He showed a good grasp in setting the
relations between potential and work, for instance in the case of the potential of
one body with respect to another193. Here in fact he precisely stated the
equivalence between potential and work. But problems occur in the definition of
the potential of a body on itself (the sum of the potentials of an electric element
of a body with respect to all the other electrical elements of the same body): in
this case Helmholtz's definition is double the modern convention, but what mostly
matters, does not correspond to the work done194 (the potential is supposed to be
double the work done). This shows the "independence" of the two concepts in
Helmholtz's approach. In the original 1847 edition of the Erhaltung there is a
final correction (the only one) referring exactly to these problems. This is a clear
indication of the incertitudes and difficulties faced by Helmholtz with concepts
that in 1847 were by no means common195. Helmholtz in 1882 196 acknowledged
190 According to Koenigsberger, Helmholtz knew of Green's theorem later, in
Koenigsberg, partially through F.Neumann: Koenigsberger H v H P.100.
Erhaltung
191 Helmholtz Erhaltung P.38.
192 Helmholtz Erhaltung P.39.
193 Helmholtz Erhaltung Pp.39-40
194 Helmholtz Erhaltung Pp.42-3. Clausius was the first to criticise this part of the
195 In the 1882 reprint of the Erhaltung in WA 1 the correction is incorporated in the
text (as in the English translations of Tyndall, Kahl and Lindsay), and Helmholtz warned the