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

conformity with Gauss in his magnetical researches" 262 as the sum of tensions
consumed by the motion from infinity to r, and, equivalently, as the sum of the vis
viva produced. Thus :
"the increase of vis viva in any movement must be considered equal to the
difference of the potential at the end of the trajectory with respect to the potential
at the beginning" 263.
That is, the sum of tension forces is equivalent to the difference of potentials:
− ∫ r
R
φ dr = e1e2 R − e1e2 r
and, of course, also to the gain in vis viva caused by passing from the distance R
to r.
Helmholtz then introduces the concept of the potential of a body in itself and of
the potential of a body on another. In a specific case given as an example, he
calculates the difference of potentials before and after the movement of a
quantity of electricity. This difference is defined as equivalent to the quantity of
work done:
-(V + (Wa +Wb)/2) 264
As already seen Clausius in 1852 criticised Helmholtz assertion that in the above
expression the potential W of a body on itself is not equal to the corresponding
work done, but twice as much (the corresponding work is in fact W/2). The
accusation of not having understood the deep relations between potential and
work was a serious one from Clausius' point of view.
Clausius in the same1852 paper had established a relation between vis viva and
potential different from that of Helmholtz's Erhaltung. He started from the vis
viva theorem and equated the increase of vis viva to the quantity of mechanical
work produced during the same time in the system 265. Clausius did not accept
Helmholtz's "sum of tension forces" (potential energy) and the corresponding
interpretation of the conservation principle. For Clausius, work, being in most
cases a total differential and thus its integral depending only on the initial and
final positions, can be identified with a difference of potentials. The potential of
an exterior system of masses on a given system is introduced as the function:
262 Helmholtz Erhaltung p. 38.
263 Helmholtz Erhaltung p. 38.
264 Helmholtz Erhaltung Pp.42-3.
265 Clausius "Electric Discharge" p.3.