new coefficients, and evidently obtain a result equivalent to equation (13) and function (12). We therefore see that the single supposition of the wave-disturbance, being always accurately in the wave's front, leads to a result equivalent to that given by the former process; and we are thus assured that by employing the simpler method we do not, in the case in question, eventually lessen the generality of our result, but merely, in effect, select the three rectangular axes, which may be called the axes of elasticity of the medium, for our co-ordinate axes. From the general form of $, it is clear that the same observation applies to it, and therefore the consequences before deduced possess all the requisite generality. The same conclusions may be obtained, whether we introduce the consideration of extraneous pressures or not, by direct calculation. In fact, when these pressures vanish, and we conceive a section of the ellipsoid whose equation is (13), made by a plane parallel to the wave's front, to turn 90 degrees in its own plane. the same reasoning oy which equation (10) was before found, immediately gives, in the present case, 1 = Lit's + 1/y? + N2 + 2 Pjé +2Qx'z' + 2Ro'y'... (14), for the equation of the surface in which all the elliptical sections in their new situations, and corresponding to every position of the wave's front, will be found. Lastly, when we introduce the consideration of extráneous pressures, it is clear, from what precedes, that we shall merely have to add to the function on the right side of the equation (13), the quantity (Aq* + Bb? + Co® + 2 Dbc + 2 Fac +2 Fab) (x + y + 2), which would arise from changing w, u, and w into a, y, and 2, Also n into a, b, c, in that part of $ which is of the second degree in u, v, w, agreeably to the remark in & foregoing note. Afterwards, when we determine the values of A, B, &c., by the same condition which enabled us to deduce which is applicable to the more general case just considered* * Vide Professor Stokes' Report on Double Refraction (British Association. 1862. p. 265). RESEARCHES ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. From the Transactions of the Royal Society & Edinburgh. [Read Dec. 16. 1833.] |