Neglecting the double integrals which relate to the extreme boundaries of the medium, and which we will suppose situated at an infinite distance, we get for the general equations of motion, If now in our indefinitely extended medium we wish to determine the laws of propagation of plane waves, we must take, to satisfy the last equations, u= af (ax + hy + cz +et), w = rf (0xx -+by t cz +et); a, b, and c being the cosines of the angles which a normal to the wave's front makes with the co-ordinate axes, A, B, y constant coefficients, and e the velocity of transmission of a wave perpendicular to its own front, and taken with a contrary sign. Substituting these values in the equations (5), and making to abridge A' =(G+A) a*-+ (N+B) 62 +(M + C) c, F" = (N+R) ab; 0 = F"a + (B – pe”) B+ Diny ... (6). 0 = E'a + D'B + (C" – pe) op? These last equations will serve to determine three values of p", and three corresponding ratios of the quantities a, B, Y; and hence we know the directions of the disturbance by which a plane wave will propagate itself without subdivision, and also the corresponding velocities of propagation. From the form of the equations (6), it is well known, that if we conceive an ellipsoid whose equation is 1 = A'm + B'yü + C'e? + 2D'yz +2E+x2 + 2Fay*......(7), • If we reflect on the connexion of the operations by which we pass from the function (4) to the equation (7), it will be easy to perceive that the right side of the equation (7) may always be immediately deduced from that portion of the function and represent its three semi-axes by r', 5”, and '", the directions of these axes will be the required directions of the disturbance, and the corresponding relocities of propagation will be given by Fresnel supposes those vibrations of the particles of the luminiferous other which affect the eye, to be accurately in the front of the wave. Let us therefore investigate the relation which must exist between our coefficients, in order to satisfy this condition for two out of our three waves, the remaining one in consequence being necessarily propagated by normal vibrations For this we may remark, that the equation of a plane parallel to the wave's front is = ax' +by' + ce ......(a) * = c+al, = 6 +ch, and substitute these values in the equation (1) of the ellipsoid; testoring the values of A', B', C', D, E, F, the odd powers of a ought to disappear in consequence of the equation (a), whatever may be the position of the wave's front. We thus get G=H=I=f suppose, and P=4-2L, second degree, by changing u, o, and w into e, y, and & Also e r den and me into a B, a This remark will be of use to us afterwards, when we come to consider the most general formi of the function due to the internal actions, IN CRYSTALLIZED MEDIA, 303 ele functie . In fact, if we substitute these values in the function (4), there will result -20=-20, -3%. which, when (= 4,0=B, 0= C, reduces to the last four lines. Making the same substitution in the equation (7), we get 1 = " (2:+by+cz) +(Aa? + Bb+ Cc) (cc* + y + ) +L (cy – bz)' + M (az - ca) + N(bx - ay)* ) Let us in the first place suppose the system free from all extraneous pressure. Then A=0; B=0, C=1, and the above equation, combined with that of a plane parallel to the wave's front, will give 0Fux + by +c&...................... (9), 1=1 (cy – bx)* + M (az – czje + N(bx -- ay)", the equations of an infinite number of ellipses which, in general, do not belong to the same curve surface. If, however, we cause cach ellipsis to turn 90° in its own plane, the whole system will belong to an ellipsoid, as may be thus shewn: Let (xyz) be the co-ordinates of any point p in its original position, and (w'y/x') the co-ordinates of the point p' which would coincide with p when the ellipsis is turned 90° in its own plane. Then and + p + z = x2 + y2 +2", since the distance from tae origin O is unaltered, 0 = arc' + by' + cé, since the plane is the same, |