Mr, Earnshaw on the Motion of Luminous Waves. 371 



fore it appears that if we would inquire into the secrets of op- 

 tical refraction as they exist in nature, we must be careful not 

 to make geometrical symmetry a necessary condition in our 

 investigations. If we neglect to observe this caution, we 

 can never be sure that some of our results may not be due to 

 our geometrical assumption : and here, perhaps, I may be al- 

 lowed to state, en passant, that after carefully weighing what 

 has been done v/ithin the last few years on the theory of the 

 transmission and refraction of light, I cannot avoid suspecting 

 that along list of theoretical results which have been obtained 

 is due to the assumption of geometrical symmetry and the mis- 

 interpretation of analytical expressions. 



In a paper printed in the Cambridge Philosophical Trans- 

 actions, vol. vii.j I have shown by a very simple analysis that 

 there exist for each particle three rectangular directions of 

 mechanical symmetry in every system of tletached particles, 

 whether those particles be arranged in geometrical symmetry 

 or not. Under certain conditions the directions of mechani- 

 cal symmetry are parallel for all particles. Now we learn 

 from experiment that these conditions are fulfilled in nature. 

 Again, experiment shows that the superposition of undula- 

 tions is an optical principle which exists in nature: on the 

 authority of this fact, I shall neglect in the equations of mo- 

 tion all powers of the displacements of the particles above the 

 first. As I shall avoid any assumption based on geometrical 

 symmetry, I shall consider my results as belonging equally to 

 crystallized and non-crystallized media. For simplicity, I 

 shall consider only the transmission of plane waves through 

 the medium in any direction. 



Let XT/ z, x' 7/' z' be the coordinates of the positions of rest 

 of an attracted particle m, and any attracting particle m' ; and 

 let 7-' be their distance. At any time t lei x + ^, y + yj, ::: + f, 

 and x' + ^', 7/' + rl, z' + f' be the coordinates, and R' the di- 

 stance of the same particles. Also let the law of attraction of 

 m' upon m be represented by m'J'{R') ; and assume F to be 

 such a function that F {u) =/f{u)du. Then the force ex- 

 erted by 7k' on m parallel to the axis of .v, at the time t, 



= ,«'/(R') . -V- -^ = - '"^(^ ) ■ ^= - '« -7^ 

 Now since 11^ = (x' - x)^ + {ij - yf + (s' - z)\ and R« 



= (^ + r - *• -jif + {y' + y)'-2/-yif +.(f' + ^-z- ^)% 



it is evident 11' is derived from R by writing x + ^ — ^',y 

 + )) — i)'> 2 + ?— ^' for 3;, j/, 2: consequently, by Taylor's 

 theorem, 



2 C2 



