tions of polarization. The two surfaces have only this section in common *, and a very simple application of our theory would shew that no force perpendicular to the wave's front is rejected, as in the ordinary one, but that the force in question is absolutely null +. Let us conceive a system composed of an immense number of particles mutually acting on each other, and moreover subjected to the influence of extraneous pressures. Then if x, y, z are the co-ordinates of any particle of this system in its primitive state, (that of equilibrium under pressure for example), the co-ordinates of the same particle at the end of the time t will become x', y', z', where x', y', z' are functions of x, y, z and t. If now we consider an element of this medium, of which the primitive form is that of a rectangular parallelopiped, whose sides are dx, dy, dz, this element in its new state will assume the form of an oblique-angled parallelopiped, the lengths of the three edges being (dx'), (dy'), (dz'), these edges being composed of the same particles which formed the three edges dr, dy, dz in the primitive state of the element. Then will (dx')' = {(dx')" + (d',')" + (de')"} do" = a*do* {(dz)* + (dzx')" + (dz')"} dx dx dy dy, dz' dz + + dy dz dy dz ay dy {(土)+(土)+(金)+(金)+(出)} dy dz dz dz [It will be seen that this remark is not strictly correct, as the surface must necessarily have another common plane section.] + [Referring to the values of u, v, w given in p. 301, we see that, since the direction of vibration is supposed to be in the front of the wave, we have Suppose now, as in a former paper, that pdx dy dz is the function due to the mutual actions of the particles which compose the element whose primitive volume = dxdy dz. Since must remain the same, when the sides (dx'), (dy'), (dz') and the cosines a, B, y of the angles of the elementary oblique-angled parallelopiped remain unchanged, its most general form must be function (a, b, c, a, B, y), = or since a, b, and c are necessarily positive, also This expression is the equivalent of the one immediately preceding, and is here adopted for the sake of introducing greater symmetry into our formulæ. We will in the first place suppose that is symmetrical with regard to three planes at right angles to each other, which we shall take as the co-ordinate planes. The condition of sym metry with respect to the plane (yz), will require to remain unchanged, when we change But thus a2, b2, c2 and a' evidently remain unaltered; moreover Applying the like reasoning to the other co-ordinate planes, we see that the ultimate result will be $=ƒ (a3, b2, c2, a”, ß'2, y'2) ................. .(2). The foregoing values are perfectly general, whatever the disturbance may be; but if we consider this disturbance as very small, we may make x = x+u, y' = y + v, z = z +20, u, v, and w being very small functions of x, y, z, and t of the first order. Then by substitution we get we thus see that s1, 8, 8, a', ß, y', are very small quantities of the first order, and that the general formula (1) by substituting the preceding values would take the form which may be expanded in a very convergent series of the form $ = 8, + $、 + é2+ &, +&c. : P., 41, 4,, &c. being homogeneous functions of 8, §, 84 à', B', 'Y', of the degrees 0, 1, 2, 3, &c. each of which is very great compared with the next following one. But being constant, if p the primitive density of the element, the general formula of Dynamics will give (d'u δυ διο dz [[[pardydz{du du + de 8v+ 21 80} = [[[d.xdyd: (84,+84,+&c.). If there were no extraneous pressures, the supposition that the primitive state was one of equilibrium would require 4,0, as was observed in a former paper; but this is not the case if we introduce the consideration of extraneous pressures. However, as in the first case, the terms p, p, &c. will be insensible and the preceding formula may be written Supposing p the primitive density constant, the most general form of o, will be ♦, = − ¦ Φι A, B, C, D, E, and F' being constant quantities. In like manner the most general form of 4, will contain twenty-one coefficients. But if we first employ the more parti cular value (2), we shall get ~ 241 = A8, + Bs,+C$§ 2 − 24,= Gs,2 + Hs22 + Is2 + 2P§ ̧§ ̧ +2 Q8,8, +2 Rs,8, + La22 + MB22 + Ný3. Or by substituting for s,, 8, 8, a', B', their values, given by system (3), continuing to neglect quantities of the third order, we get Having thus the form of the function due to the internal actions of the particles, we have merely to substitute it in the general formula of Dynamics, and to effect the integrations by parts, agreeably to the method of Lagrange. Thus, |