476 THE LE CHATELIER-BRAUN PRINCIPLE [418
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. 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 dimmish this descent."
It may suffice here to sketch the demonstration for the case of two degrees of freedom, the results of which may, indeed, be interpreted so as to cover most of the ground. The potential energy of the system, slightly displaced from stable equilibrium at « = 0, y = Q, may be expressed
where, in virtue of the stability, a, c, and ac — I- are positive. The forces X, Y, corresponding with the displacements x, y, and necessary to maintain. these displacements, are :
X = d Vfdac — ax + by, Y=d V/dy - bx + cy. If only X act, that is, if Y— 0, y = — bx/c, and
X
00 =
a - bz/c'
This is the case of no constraint. On the other hand, if y is constrained to remain zero by the application of a suitable force F, the relation between the new x (say #') and X is simply
, X
x = —. a
T' /)2
Thus - - = !--;
as ac
so that #', having the same sign as x, is numerically less, or the effect of the constraint is to diminish the displacement as due to the force X. An exception occurs if b = 0, when x = X/a, whatever y and F may be, so that the constraint has no effect.
An example, mentioned by Ehrenfest, may be taken from a cylindrical rod of elastic material subject to a longitudinal pressure, X, by which the length is shortened (as). In the first case the curved wall is free, and in the second the radius is prevented from changing by the application of a suitable pressure. The theorem asserts thafc in the second case the shortening due to the longitudinal pressure X is less, in virtue of the constraint applied to the
walls.
i
Returning to the compressed gas, we now recognise that it is the pressure &p which is the force and — 8v the effect, corresponding respectively with X and x of the general theorem. But we may still feel a doubt as to which is the constrained condition, the isothermal or the adiabatic, and without a decision on this point no statement can be made. It is, however, evident that if the general theorem is applicable at all, the adiabatic condition mustlevelling stand. Nothing particular happened afterwards for days or weeks; but eventually parts of the gelatine film lifted, carrying up with them material torn away from the glass. The plate is still in my possession, and there is now but little of the original glass surface left. If the process is in regular use, I should much like to know the precise procedure. It seems rather mysterious that a film of gelatine, scarcely thicker than thick paper, should be able to tear out fragments of-solid glass, but there is no doubt of the fact.