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

point: as far as a potential exists, the corresponding force law does not contradict
the impossibility of perpetual motion. Since this is the first root of the principle of
conservation of vis viva analysed in section 3, it is to be asserted that Weber’s
law originates directly from this root. This approach to PCE was often called the
law of potential 394.
Weber in 1871 proposed his own version of the principle of energy
conservation 395. At the end of the second part of his 1871 paper Weber answers
Helmholtz’s criticisms of 1870 about the incompatibility of Weber’s force law
with some consequences of PCE. This was the starting point of the second part of
the lengthy controversy. Weber first notes the new requirement Helmholtz
imposes on PCE. The vis viva cannot become infinite, or otherwise an infinitely
great amount of work could be performed (either in passing from a finite velocity
to an infinite, or from an infinite to a finite). Thus a limiting velocity must exist.
For Weber this limiting velocity is his constant cw=⎟2 vel.light. In Weber’s
view, Helmholtz’s criticism has to be rejected for it assumes an initial velocity of
the particles greater than c. Assuming this limiting value Weber’s potential is
always a positive quantity. In the second place, Weber notes, the finite distance
at which the particles in Helmholtz’s objection would acquire an infinite velocity
is extremely small, outside the domain of enquiry. Thus the objection is
practically meaningless. But Helmholtz again in 1872 and 1873 criticised
Weber’s law because of the negative sign in one of the terms of the generalised
potential. In fact this implies that a charge behaves "somewhat as if its mass were
negative, so that in certain circumstances its velocity might increase indefinitely
under the action of a force opposed to the motion" 396. Maxwell in 1873 agreed
with Helmholtz’s 1872 criticisms and asserted that the latter "... impossible result
is a necessary consequence of assuming any formula for the potential which
introduces negative terms into the coefficient of v2" 397. Hoppe, an historian
who defended Weber’s assumptions in two books, first in 1884 and then in 1927,
asserts that in 1875 Weber succeeded in showing that with a proper
reinterpretation of his equation for the vis viva, Helmholtz’s criticism can be
394 Neumann Amp u Web p.337.
395 Weber, Wilhelm. "Elektrodynamische Maassbestimmungen, insbesonder uuber das
Prinzip der Erhaltung der Energie." 1871. In Werke 6 vols. Berlin, 1892-4. Vol.4. Pp. 247-99.
Tr. in Phil Mag 43 (1872): 1-20 and 119-49.
396 Whittaker Aether p.204.
397 Maxwell Treatise par.854.