IN CRYSTALLIZED MEDIA. 139 



But if in equation (8) and corresponding function (A), we suppose 

 A = 0, B = and C = 0, and then refer the equation to axes taken 

 arbitrarily in space, we shall thus introduce three 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 elas- 

 ticity of the medium for our co-ordinate axes. From the general form 

 of 0, 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 by which equation 

 (10) was before found, immediately gives, in the present case, 



1 = Lx' 2 + My' 2 + Nz' 2 + 2 Py'z' + 2 Qx'z + 2 Ra'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 extraneous 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 



(Aa 2 + Bb 2 + Cc 2 + 2Dbc + 2Eac + 2 Fab) (x 2 + f + z 2 ), 

 which would arise from changing u, v and w into x, y and as. Also -r- , -r- , 



-j- into a, b, c, in that part of (p x which is of the second degree in u, v, w, 

 agreeably to the remark in a foregoing note. Afterwards, when we 



S2 



