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

directions that join them and whose intensity depends only on the distance, that is
only for central forces. But Clausius claims that the demonstration given by
Helmholtz does not justify the assertion.
In fact, during the demonstration, one of the very statements to be derived, that
the intensity of the force is a function of the distance, is presupposed 281. This
presupposition is part of Helmholtz's model as expressed in the Einleitung. Thus
the principle of vis viva is either unnecessary to derive the central forces (if we
already presuppose that the intensity is a function of the distance) or is
insufficient (if we do not make that assumption).
Moreover, Clausius asserted that from the vis viva theorem, assuming that the
vis viva is a function only of the space coordinates, we can derive that the work
too is a total differential and also that the force acting is a function of the space
coordinates. But this does not imply that the force be central. The only possible
implication for Clausius is that, if one of the two conditions hold (direction or
intensity) the other does too.
Finally Clausius, while not questioning the "physical probability", denies the
"mathematical necessity" of central forces 282.
Helmholtz, apart from deriving them from the vis viva principle in the second
chapter, had assumed in the Introduction that a point acts in different directions
with the same force. Clausius does not consider this as evident: the contrary is
not "unthinkable". Moreover in Helmholtz's first chapter of the Erhaltung, when
the principle of virtual velocities is derived from the vis viva principle it is also
asserted that the result is that the forces of two points act on the direction joining
them. But for Clausius this, again, was part of the assumptions and cannot be
considered a deduction. Helmholtz's whole attempt to demonstrate the
mathematical need for central forces is, for Clausius, a flaw.
The core of Helmholtz's program was seriously shaken. No doubt he was deeply
worried by the criticisms for he dedicated a great effort to preparing a twenty
page paper, to be published in 1854 and reprinted in 1882 in the energy section
of the WA 1, after the Erhaltung.
Replying to Clausius in this paper, Helmholtz asserted that he could derive from
the vis viva principle the need for the forces to be central, without further
hypotheses, because he expressed the principle only with reference to the relative
positions of the points and not to the absolute ones. In Helmholtz's view this
281 Clausius "Einige Stellen" p.575.
282 Clausius "Einige Stellen" p.577.