the triple integrals extending over the whole volume of the medium under consideration.

But by the supposition, when u=0, v=0 and w = 0, the system is in equilibrium, and hence

0 = [[fdx dy dz 84,

seeing that, is a homogeneous function of 8,, 3, 8, u, B, y of the first degree only. If therefore we neglect 4, 4, &c. which are exceedingly small compared with 4,, our equation becomes

the integrals extending over the whole volume under consideration. The formula just found is true for any number of media comprised in this volume, provided the whole system be perfectly free from all extraneous forces, and subject only to its own molecular actions.

If now we can obtain the value of 4,, we shall only have to apply the general methods given in the Mécanique Analytique. But being a homogeneous function of six quantities of the second degree, will in its most general form contain 21 arbitrary coefficients. The proper value to be assigned to each will of course depend on the internal constitution of the medium. If, however, the medium be a non-crystallized one, the form of will remain the same, whatever be the directions of the co-ordinate axes in space. Applying this last consideration, we shall find that the most general form of 4, for non-crystallized bodies contains only two arbitrary coefficients. In fact, by neglecting quantities of the higher orders, it is easy to perceive that

and if the medium is symmetrical with regard to the plane (xy) only, will remain unchanged when -z and -w are written for z and w. But this alteration evidently changes a and 8 to

a and B. Similar observations apply to the planes (x) (y). If therefore the medium is merely symmetrical with respect to each of the three co-ordinate planes, we see that must remain unaltered when

In this way the 21 coefficients are reduced to 9, and the resulting function is of the form

Probably the function just obtained may belong to those crystals which have three axes of elasticity at right angles to each other.

Suppose now we further restrict the generality of our function by making it symmetrical all round one axis, as that of z for instance. By shifting the axis of x through the infinitely small angie de,

Making these substitutions in (4), we sec that the form of 4, will not remain the same for the new axes, unless

under which form it may possibly be applied to uniaxal crystals.

Lastly, if we suppose the function 4, symmetrical with respect to all three axes, there results

or, by merely changing the two constants and restoring the values of a, B, and Y,

This is the most general form that 4, can take for non-crysallized bodies, in which it is perfectly indifferent in what directions the rectangular axes are placed. The same result might be obtained from the most general value of 4,, by the method before used to make 4, symmetrical all round the axis of z, applied also to the other two axes. It was, indeed, thus I first obtained it. The method given in the text, however, and which is very similar to one used by M. Cauchy, is not only more simple, but has the advantage of furnishing two intermediate results, which may possibly be of use on some future occasion.

Let us now consider the particular case of two indefinitely extended media, the surface of junction when in equilibrium being a plane of infinite extent, horizontal (suppose), and which we shall take as that of (yz), and conceive the axis of a positive directed downwards. Then if p be the constant density of the upper, and p, that of the lower medium, 4, and 4," the corresponding functions due to the molecular actions; the equation (2) adapted to the present case will become

u, v,, w, belonging to the lower fluid, and the triple integrals being extended over the whole volume of the fluids to which they respectively belong.

(1)

It now only remains to substitute for 4, and their values, to effect the integrations by parts, and to equate separately to zero the coefficients of the independent variations. Substituting therefore for 4, its value (C), we get

jdu dv

du dw
dz dx.

(3x + 1) 8}

800

d'u d'u d (dv

+ + +B +

dx dx dy dz [dy dz* dx dy

+

d jdu dv

dw

dz.

dx

]}

80

Ev

διο ;

>

dy dz Xx

dy

seeing that we may neglect the double integrals at the limits
x=-∞, y = ±∞, ∞; as the conditions imposed at these
limits cannot affect the motion of the system at any finite dis-
tance from the origin; and thus the double integrals belong
only to the surface of junction, of which the equation, in a state
of equilibrium, is

0 = X.

Su