B. Relations between forces and velocities.

It follows further from equations (7) that:

Thus: (9b)

In the very great number of cases, where: (9c)

it will follow that: (9d)

i.e., when a rise in the velocity qb for the same position and acceleration makes the force Pa increase, a corresponding rise in qa will diminish the force Pb . The case in which the prerequisite (9c ) is fulfilled have already been remarked in the examples that were cited in A.

They best show the extended meaning of this theorem, but also the fact that one must control the fulfillment of the prerequisite, before one applies the simpler theorem (9d), instead of the generally correct one (9b ).

Example I. If a force that increases the angle β – i.e., the axis of the top tends to move from the vertical – causes a greater precessional motion α then a force that causes the precessional motion to accelerate will bring the axis to the vertical line.

Example II. Electromagnetic induction, according to Lenz. The motion of two circular currents with respect to each other that is produced by ponderomotive, electrodynamical forces will bring about electromotive, induced forces that act against the currents.

The corresponding relationship will be true for the motion of a magnet relative to a current conductor.

Example III. Thermodynamics.

When rises in temperature raise the pressure of a system of bodies, compression of them will raise the temperature. For this case, we can write equation (9d ), after multiplying both sides by η, using the notations and explanations of § 2 for this example: (9e)

Now, from (6c), one has:

Thus, one has: (9f)

From (6d), one had:

and since L is independent of h, one will have: (9g)

by which, in conjunction with (9 f), the validity of equation (9c ) will be confirmed, and thus, also the applicability of our general theorem. Thus, any of the functions η in equation (6b ) can be regarded as the velocity, except that dη / dt must then correspondingly mean the acceleration. Also, the temperature ϑ, in turn, belongs to the integrating denominator η such that one also has:

Since one must have dϑ / dt = 0 in this application, dQ

q dt ∂     ∂ a   will be the velocity with which the heat enters when the parameter pa increases with the velocity, while ϑ remains constant.

This will give the formulation of the theorem that was given above. The same considerations can also be applied to the reversible parts of thermoelectric and electrochemical processes. Peltier’s phenomenon: If warming at one place in a closed conductor brings about an electrical current then the same current will produce cooling there (ignoring the formation of heat by the resistance of the conductor.) Electrochemistry: If warming of a constant galvanic element raises the electromotive force then the current in it will make the heat latent ( * ).

However, the formulas above not only exhibit the sense of the change, but also, at the same time, they give one information about how one is to deal with the quantities.