459 



it may be shown that the function Fis so related to the two 

 sets of axes of tliese ellipsoids, that by assunaing either set of 

 axes for axes of coordinates, we may destroy three coeflBcients 

 of the function. 



" Assuming as co-ordinate axes, the axes of the second of 

 ellipsoids (G), the equation of the surface of wave-slowness is 

 found to be the following : 



[[(x^ + 2/2 + 2^) {QRx'' + PRy^ + PQz")" 

 {E- 1) ^+(£;-l){(Q+2^)a;2+(P+J?)y2+(P+Q)22) (7) 



where 



E = A3?\By^\ Cz^ + IFyz + 'iGxz + 2Hxy. 



" The equation is thus seen to be composed of two factors; 

 the first, of the fourth degree, representing a surface with two 

 sheets, which belong to the twotransverse vibrations; thesecond, 

 of the second degree, representing an ellipsoid which belon^-s 

 to the normal vibration. The surface of the fourth degree 

 may be shown to have sixteen multiple points, and conse- 

 quently, its reciprocal polar, or wave surface, will be reduced 

 from the thirty-sixth to the fourth degree. 



" In the particular case in which the normal vibration 

 vanishes, we shall have £J = 0, and the surface of transverse 

 vibrations will become Fresnel's wave surface, as is evident 

 on inspection of equation (7). The function corresponding 

 to this reduction will be 



2 F = P ( X^ - Ml) + Q ( y^ - y,) + iE (Z2 - wz). (8) 



" In the last section of the memoir published in vol. xxii. 

 Part I. of the Transactions of the Royal Irish Academy, I 

 have directed attention to the fact, that the deduction of the 

 laws of wave propagation is no proof of the truth of any me- 

 chanical theory of light, as this deduction may be made from 

 several theories. There are, in fact, no less than five distinct 



