b Mn GREEN, ON THE REFLEXION 



But by the supposition, when u = 0, v = and w = 0, the system is 

 in equilibrium, and hence 



= fffdx dy da $fa : 



seeing that 0, is a homogeneous function of s lf #*, s 3 , a, /3, 7 of the 

 , first degree only. If therefore we neglect fa, fa, &c. which are ex- 

 ceedingly small compared with <p 2 , our equation becomes 



fffp dx dy dz i^ iu + ( -~ It, + -^ SwJ = fffdx dy dz Sfa (2) ; 



the integrals extending over the whole volume under consideration. 

 The formula just found is true for any number of media comprised in 

 this volume, provided the whole system be perfectly free from all 

 extraneous forces, and subject only to its own molecular actions. 



If now we can obtain the value of fa, Ave shall only have to apply 

 the general methods given in the Mecaniquc Analytique. But fa, being 

 a homogeneous function of six quantities of the second degree, will in 

 its most general form contain 21 arbitrary coefficients. The proper 

 value to be assigned to each, will of course depend on the internal 

 constitution of the medium. If, however, the medium be a non-crystal- 

 lized one, the form of fa will remain the same, whatever be the 

 directions of the co-ordinate axes in space. Applying this last con- 

 sideration, we shall find that the most general form of fa for non- 

 crystallized bodies contains only two arbitrary coefficients. In fact, by 

 neglecting quantities of the higher orders, it is easy to perceive that 



_ du _ dv dw 



ax dy dz 



dw dv _ div du du dv 



dy dz' dx dz' T'T" dy dx' 



and if the medium is symmetrical with regard to the plane (xy) only, 

 fa will remain unchanged when — z and — to are written for z and w. 

 But this alteration evidently changes a and j8 to — a and — /3. Similar 

 observations apply to the planes (xz) (yz), If therefore the medium is 

 merely symmetrical with respect to each of the three co-ordinate planes, 

 we see that fa must remain unaltered when 



