310 ON THE PROPAGATION OF LIGHT 



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 accu- 

 rately 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 intro- 

 duce the consideration of extraneous pressures or not, by direct 

 calculation. In fact, when these pressures vanish, and we con- 

 ceive 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'* + My* + Nz* + ZPy'z + 2Qa?V + ZEx'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 



(Ac? + BV + Cc* + 2Dbc + 2Eac + 2Fab) (x z 

 which would arise from changing u, v, and w into x, y, and z. 

 Also -y- , -y- , -=- into a, 6, c, in that part of < which is of 



the second degree in u, v, w, agreeably to the remark in a 

 foregoing note. Afterwards, when we determine the values of 

 A, B, &c., by the same condition which enabled us to deduce 



