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the forces in nature are so disposed as to render this a natural impossibility.

Let us now take any element of the medium, rectangular in a state of repose, and of which the sides are dx, dy. dz\ the length of the sides composed of the same particles will iu a state of motion become

dx-dx (1 +«,), dy' = dy (l+*»), dz' = dz (1 +«„);

where s», s, are exceedingly small quantities of the first order. If, moreover, we make,

«-«»<%, 0-«»<%, 7=cos<|:;

a, ft, and 7 will be very small quantities of the same order. But, whatever may be the nature of the internal actions, if we represent by

Stj) dx dy dz,

the part of the second member of the equation (1), due to the molecules in the clement under consideration, it is evident, that if> will remain the same when all the sides and all the angles of the parallelopiped, whose sides are dx dy dz, remain unaltered, and therefore its most general value must be of the form

^ = function {*,, s„ s„ a, ft, 7].

But s,, 8lt ss, a, ft, 7 being very small quantities of the first order, we may expand <ff > in a very convergent series of the form

+ & + +

fa, fa, fa, &c. being homogeneous functions of the six quantities a, ft, 7, st, st, s, of the degrees 0, 1, 2, &c. each of which is very great compared with the next following one. If now, p represent the primitive density of the element dx dy dz, we may write p dxdy dz in the place of Dm in the formula (1), which will thus become, since fa is const.TM*.

//fp ** dy Zu+Ib dv+2?Bw. .

- jj jdx dy dz + + &c.);

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

But by the supposition, when u = 0, V = 0 and to = 0, the system is in equilibrium, and hence

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seeing that is a homogeneous function of s,, sa, a, /3, y of the jtrst degree only. If therefore we neglect <&,, &c. which are exceedingly small compared with £a, our equation becomes

jjjpdxdydz +^3 ^v= jjjdxfydzHt""(i),

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 we shall only have to apply the general methods given in the MJcanique 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 <f>t for non-crystallized bodies contains only two arbitrary coefficients. In fact, by neglecting quantities of the higher orders, it is easy to perceive that

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and if the medium is symmetrical with regard to the plane (xy) only, <f>2 -will remain unchanged when — z and — w are written for z and w. But this alteration evidently changes a and ft to — a and — ft. Similar observations apply to the planes (xz) (yz). If therefore the medium is merely symmetrical with respect to each of the three co-ordinate planes, we see that <f>t must remain unaltered when

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

+ 2P^V ^W + 20^U ^W 222 ^U ^V d> (A) dy'dz dx'dz dx'dy '*

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 S6f

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

G = H=<2N+R,

L = M,

P=Q;

and thus we get

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

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

G = H=I=2N+R,

L = M=N,

P=Q=R;

and consequently,

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or, by merely changing the two constants and restoring the values of a, £, and 7,

~.' i fdu dv dw\*

(\</y dx) \dz dx) \dz dy)

^ fdv dw ^du dw du dv\ .~

\dy' dz dx' dz dx' dy))

This is the most general form that <f>s can take for n6n-crysaillized 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 $t, by the method before used to make ^, 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 x positive directed downwards. Then if p be the constant density of the npper, and p, that of the lower medium, <pt and <pTM the corresponding functions due to the molecular actions; the equation (2) adapted to the present case will become

+ /fjP.dxdydz J-3 Su,+g< Sv, + j ,

•zjjjdxdydzh+jjjdxdydzti" (3);

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

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