IN THE INTERIOR OF TRANSPARENT BODIES. 433 



matter : hence, P, Q, R, S, C recur when we pass from one particle to 

 another similarly situated with respect to the particles of matters. 



$ 41. It will be necessary to recur to the original equations (B), 

 in which I shall suppose the particles of matter fixed, and therefore 



«, = 0, $ym 0, 7/ = 0. 



Let us put a = a + e, /3 = /3 + »/, 7 = 7 + £; when, a, ft, 7 are 

 displacements, such as constitute a common wave of light in vacuum, 

 and e, tj, £ the quantities to be added to them in order to satisfy the 

 equations (B). Then, denoting the second member of (2?) by F(a ft 7), 

 we have 



w+£»wfl +■*<••.& 



Now since a ft 7 are displacements, such as constitute a common 

 wave of light in vacuum, we may expand F (a ft 7) as in $ (8), 

 neglecting all above second differential coefficients. In the part retained 

 the differential coefficients will be multiplied not by constant quantities, 

 but by periodical coefficients, functions of # y %, which recur in the man- 

 ner just described. Let F x (a ft 7) be the value of F (a ft 7), when we 

 put for these coefficients their mean values, and let F z (a /3 7) be the 

 value of F(a (3 7), when we omit the mean part of each coefficient, 

 and retain only its periodical part, 



then, F(a p 7) = F l (o ft 7) + F 2 (a ft 7). 



In F, (a ft 7) the differential coefficients are multiplied by constant quan- 

 tities ; and in F s (a ~ft 7) they are multiplied by periodical functions which 

 go through all their values when we pass from one particle of matter 

 to another similarly situated with respect to the particles of matter, and 

 whose mean values are zero. 



Substituting this value of F (a ft 7), the equations {B) become 



^ + ^ = FCafty) + F 2 (afty) + F( ev 0, 

 and similar equations with reference to the axes of y and x. 



