AND REFRACTION OF LIGHT. 5 



Let us now take any element of the medium, rectangular in a state 

 of repose, and of which the sides are dx, dy, dz the length of the 

 sides composed of the same particles will in a state of motion become 



dx' — dx (1 + Si), dy'— dy(\ + s 2 ), dz' '■= d*(l + # 3 ) 5 



where s t , s 2 , s 3 are exceedingly small quantities of the first order. If, 

 moreover, we make 



dy' „ dx' dx 



a = cos<^,, fi-vm< Mt 7 = cos< rfy ; 



a, fi and 7 will be very small quantities of the same order. But, what- 

 ever may be the nature of the internal actions, if we represent by 



cS(p dx dy dx, 



the part of the second member of the equation (1), due to the molecules 

 in the element under consideration, it is evident, that <p will remain the 

 same when all the sides and all the angles of the parallelopiped, whose 

 sides are dx dy' dz', remain unaltered, and therefore its most general 

 value must be of the form 



(p = function \t u s if s 3 , a, fi, y}. 



But 8 U s 2 , s 3 , a, /3, 7 being very small quantities of the first order, 

 we may expand cp in a very convergent series of the form 



(p = (p + </>, + (p 2 + 03 + &c. : 



<po, <pi, &> &c. being homogeneous functions of the six quantities 

 a, fi, 7, g u s if s 3 of the degrees 0, 1, 2, &c. each of which is very great 

 compared with the next following one. If now, p represent the 

 primitive density of the element dx dy dz, we may write p dx dy dz 

 in the place of Dm in the formula (1), which will thus become, since 

 <p is constant, 



fffp dx dy dn^Su + ^U + tLg j w J 



= fffdx dy dz (50, + $(p 2 + &c.) ; 



the triple integrals extending over the whole volume of the medium 

 under consideration. 



