*', y', z' denote any other velocities consistent with the connexions of the system, the Principle of Virtual Velocities gives Σ { (P—mx) x' + (Q—my)ÿ' + (R—mž) ¿'} =0, . (1) by means of which the initial velocities x &c. are completely determined.

In equation (1) the hypothetical velocities &c. are any whatever consistent with the constitution of the system; but if they are limited to be such as the system could acquire under the operation of the given impulses with the assistance of mere constraints, we have

2T'=Σm(x12+y12 +¿12) =Σ (Px' +Qy' + R¿')..

(2)

This includes the case of the actual motion. Returning now to the general case, suppose that E' denotes the function

E' =Σ(Px' + Qy' + R¿') — § Σm(¿12 +ÿ12 +¿12),

becoming for the actual motion

-

(3)

E=T={Σm(x2 +ÿ2 +¿2),

(4)

or, for any motion of the kind considered in (2),

For the difference between E' and E in (3) and (4), we get

E−E'= {Σm(x2+ y2 + ¿2) + †Σm(¿12 +ÿ12 +¿12)
−Σ(Px'+Qÿ'+Rż'),

in which by (1),

so that

Σ(P ''+Qÿ'+R¿') =Σm(a x' + ÿ ÿ' + ż ż1) ;

E — E' = § Σm { (x — ¿1)2 + (ÿ —¿1)2 + (ż—¿1)2},・ ・ (6) which shows that E is a maximum for the actual motion (in which case it is equal to T), and exceeds any other value E' by the energy of the difference of the real and hypothetical motions. From this we obtain Bertrand's theorem, if we introduce the further limitation that the hypothetical motion is such as the system can be guided to take by mere constraints; for then by (5)

E-E-T-T.

By means of the general theorem (6) we may prove that the energy due to given impulses is increased by any diminution (however local) in the inertia of the system. For whatever the motion acquired by the altered system may be, the value of E

corresponding thereto (viz. T) is greater than if the velocities had remained unchanged; and this, again, is evidently greater than the actual E (viz. T) of the original motion. The total increase of energy is equal to the decrease which the alteration of mass entails in the energy of the former motion, together with the energy (under the new conditions of the system) of the difference of the old and new motions. If the change be small, the latter part is of the second order.

On the other hand, of course, any addition to the mass lessens the effect of the given impulses.

A similar deduction may be made from Thomson's theorem, which stands in remarkable contrast with that above demonstrated. The theorem is, that if a system be set in motion with prescribed velocities by means of applied forces of corresponding types, the whole energy of the motion is less than that of any other motion fulfilling the prescribed velocity-conditions. "And the excess of the energy of any other such motion, above that of the actual motion, is equal to the energy of the motion which would be generated by the action alone of the impulse which, if compounded with the impulse producing the actual motion, would produce this other supposed motion." From this it follows readily that, with given velocity-conditions, the energy of the initial motion of a system rises and falls according as the inertia of the system is increased or diminished*.

We now pass to the investigation of some statical theorems which stand in near relation to the results we have just been considering. The analogy is so close that the one set of theorems may be derived from the other almost mechanically by the substitution of "force" for "impulse," and "potential energy of deformation" for "kinetic energy of motion." A similar mode of demonstration might be used but it will be rather more convenient to employ generalized coordinates.

Consider then a system slightly displaced by given forces from a position of stable equilibrium, from which configuration the coordinates are reckoned. The potential energy of the displacement V is a quadratic function of the coordinates 1, &c.

where Y, Y, &c. are the forces, E' will be an absolute maximum for the position actually assumed by the system. In equation (8), V is to be understood merely as an abbreviation

* See a paper by the author on Resonance, Phil. Trans. 1871, p. 94.

for the right-hand member of (7), and the displacements, &c. are any whatever.

In the position of equilibrium, since then Y,=

dV

&c.,

(9)

and thus

E— E' = } (¥1Y1 + . . .) + V' — (¥‚†, +.....)

Now, by a reciprocal property readily proved*,

1

where ', &c. is the set of forces necessary to maintain the configuration, &c. Thus by (10) and (11),

E—E' = } (¥‚—YT'1) (†, −Y',) + ··.,

(12)

a positive quantity representing the potential energy of the deformation (-1) &c. Thus E' attains its greatest value E in the case of the actual configuration, and the excess of this value E over any other is the potential energy of the displacement which must be compounded with either to produce the other. So far the displacement represented by &c. is any whatever; but if we confine ourselves to displacements due to the given forces and differing from the actual displacements only by reason of the introduction of constraints limiting the freedom of the system, then E'-V'; and the theorem as to the maximum value of E' may be stated with the substitution of V' for E'. Thus the introduction of a constraint has the effect of diminishing the potential energy of deformation of a system acted on by given forces; and the amount of the diminution is the potential energy of the difference of the deformations.†

For an example take the case of a horizontal rod clamped at one end and free at the other, from which a weight may be suspended at the point Q. If a constraint is applied holding a point P of the rod in its place (e. g. by a support situated under it), the potential energy of the bending due to the weight at Q is less than it would be without the constraint by the potential energy of the difference of the deformations. dV' &c.

* By substituting Y1 =

dV dy

&c., Y'1 =

dy'

1

† Compare Maxwell's Theory of Heat,' p. 131.

And since the potential energy in either case is proportional to the descent of the point Q, we see that the effect of the constraint is to diminish this descent.

The theorem under consideration may be placed in a clearer light by the following interpretation of the function E.

In forming the conditions of equilibrium, we are only concerned with the forces which act upon the system when in that position; but we may, if we choose, attribute any consistent values to the forces for other positions. Suppose, then, that the forces are constant, as if produced by weights. Then, in any position, E denotes the work, positive or negative, which must be done upon the system in order to bring it into the configuration defined by V=0. Thus, to return to the rod with the weight suspended from Q, E represents the work which must be done in order to bring the rod from the configuration to which E refers into the horizontal position. And this work is the difference between the work necessary to raise the weight and that gained during the unbending of the rod. Further, if the configuration in question is one of equilibrium with or without the assistance of a constraint (such as the support at P), the work gained during the unbending is exactly the half of that required to raise the weight; so that E is the same as the potential energy of the bending, or half the work required to raise the weight.

po

When the rod, unsupported at P, is bent by the weight at Q, the point P drops. The energy of the bending is the same as the total work required to restore the rod to a horizontal sition. Now this restoration may be effected in two steps. We may first, by a force applied at P, raise that point into its proper position, a process requiring the expenditure of work. The system will now be in the same condition as that in which it would have been found if the point P had been originally supported; and therefore it requires less work to restore the configuration VO when the system is under constraint than when it is free. Accordingly the potential energy of deformation is also less in the former case.

We may now prove that any relaxation in the stiffness of a system equilibrated by given forces is attended by an increase in the potential energy of deformation. For if the original configuration be maintained, E will be greater than before, in consequence of the diminution in the energy of a given deformation. A fortiori, therefore, will E be greater when the system adjusts itself to equilibrium, when the value of E is as great as possible. Conversely, any increase in V as a function of the coordinates entails a diminution in the actual value of V corresponding to equilibrium. Since a loss of freedom may be

regarded as an increase of stiffness, we see again how it is that the introduction of a constraint diminishes V.

The statical analogue of Thomson's theorem for initial motions. refers to systems in which given deformations are produced by the necessary forces of corresponding types-for example, the rod of our former illustration, of which the point P is displaced through a given distance, as might be done by raising the support situated under it. The theorem is to the effect that the potential energy V of a system so displaced and in equilibrium is as small as it can be under the circumstances, and that the energy of any other configuration exceeds this by the energy of that configuration which is the difference of the two.

To prove this, suppose that the conditions are that 1, ,...., are given, while the forces of the remaining types Tr+1, +2, &c. vanish. The symbols 1, &c., Y,, &c. refer to the actual equilibrium-configuration, and y,+Ayı, Y2+A¥2, &c., Y1 +▲Ý1, Y+AY, &c. to any other configuration subject to the same displacement-conditions. For each suffix, therefore, either A or vanishes. Now for the potential energy of the hypothetical deformation we have

1

2(V+AV)=(Y1+AY1) (Y1 + A¥1) +....

=2V+Y‚AY,+Ÿ¿A¥2+....

+ΔΥ. +ΔΥ...+....

...

Ψ. Δψ +2. Δύο+... =ΔΨ. Ψι+ΔΨ.Ψ.+...,

of which the former by hypothesis is zero. Thus

2AV=AY, . A¥1+A¥1⁄2· A¥2+........

as was to be proved.

2

(14)

The effect of a relaxation in stiffness must clearly be to diminish V; for such a diminution would ensue if the configuration remained unaltered, and therefore still more when the system returns to equilibrium under the altered conditions. It will be understood that in particular cases the diminution spoken of may vanish.

The connexion between the two statical theorems, dealing respectively with systems subject to given displacements and systems displaced by given forces, will be perhaps brought out more clearly by another demonstration of the latter. We have to show that the removal of a constraint is attended by an increase in the potential energy of deformation. By a suitable choice of coordinates the conditions of constraint may be