of Luminmis Waves m Transparent Bodies. 207 



except therefore in these cases our equations will have va- 

 riable coefficients, and it will be impossible to solve them in 

 general. 



But if we suppose that the particles of eether are arranged 

 as in figures 1, 2, or 3, then we may evidently take the axes 

 of coordinates in such positions that all the particles both 

 of matter and gether shall be similarly circumstanced with re- 

 spect to them : for instance, supposing the particles of matter 

 to be arranged at the corners of cubes, and therefore the par- 

 ticles of aether at the centres of these cubes, it is evident that 

 if the coordinate axes be assumed parallel to the edges of these 

 cubes, then the particles will be similarly circumstanced with 

 respect to each axis. 



Proceeding then upon the supposition that the particles are 

 arranged as in figures 1, 2, or 3, we may assume that the axes 

 of coordinates are so chosen that the particles are similarly 

 circumstanced with respect to each of them. 



Now, this being the case, it is evident that the quantity 

 S [m X any function of r x the product of any powers of 8 ar, 

 Sj/s ^ ^i) will be zero, unless each of the powers of 8 x, ly, 8 z 

 be even; and we may interchange Ix^yhz in this sum with- 

 out altering its value. The same is evidently true with re- 

 spect to the sums 



S [m^y. function of r' x powers oix — x, y,—y, ^;— 2} 



S {ni X function of r'x powers o( x—x,y—yi ~— ~J 

 and Z' {tHiX function of r^ x powersof S ^^8^^8 ;:'y}. 



Moreover, all these sums are constants, independent of x 

 y z, or Xiy^ z^. Hence if we put 



M = 2mf[r)lx'^ N = Sm—f'{r) 8a?2 8/ 



F = Sm 



r 



i/'MS^" C= im.^cf (.•') +i^-V(,•')(.^. _^.)2|, 



our differential equation becomes 



(Pa _ M^((Pa^ d^ d^\ P^ d^ 



N /d^« d^u ^ d^Q ^ d'^y \ 



^ 2 \dy^ ^ dz^ ^ dxdy ^ dxdz) 



f differential coefficients of the fourth and higher orders, 



- C « + ^?«, ^ (;•') «^ + ^V 4>' ('•') {[oc^-xf «, 



+ (•^■/-•J^) iy-y) A + (j-/-^) (-/-2)y/) J-. 



Let us now compare the relative magnitudes of tlie terms 



