M. Lorenz on the Theory of Light. 213 



still subsist among the coefficients a, the coefficients #, on the 

 other hand, will satisfy the general equation 



1>Px<i=—Kp> ( 13 ) 



or 



#1,2= — #2,1 ) #1,3=— #3,1 ) #2,3= — #3,2 t 



#j,i = 0; #2,2=0; #3,3=0. 



Uneven powers only of a p will occur in these coefficients ; they 

 are accordingly small in comparison with the coefficients a, and 

 will disappear in the first approximation (^=0), and hence also, 

 upon our present supposition, the results of the previous section 

 remain unchanged for the first approximation. 



I have carried out the calculation for the two cases of three or 

 four of the quantities p p giving a constant sum ; the latter case, 

 however, I have treated only upon the hypothesis that the quan- 

 tities e p are very small. But since the results already given 

 are always the same, I take the liberty of limiting the demon- 

 stration to the following case, 



Pi+P* + Pa=*> (14) 



where A is a constant. It will also be assumed that both e p and 

 u p are small quantities. 



While equation (4) still holds, an alteration will now, on the 

 contrary, occur in equation (6) every time that the index p has 

 one of the values 1, 2, or 3. Thus, if we look for the coeffi- 

 cients of C(pj) in the differential equations (A), we find for^ = l 

 on the left of equation (6), 



10.)+ |?o+ [fK-ft) + | f(-p 2 )]cosA 



-[|!'(-ft)+fl'(-p 2 )]sinA. . . . (15) 



If we neglect the second and higher powers of ct p , we get from 

 equation (6) for all values of p, 



ffe) =v(p P ) = IM = r( Pp ) 



l p rrip n p ^//2 + m 2 + ?2 2 



. . . (16) 



P • P ■ P 



We will further introduce into these equations the following 

 symbols, 



h =l ~~7T> ™p=m~-r, n- = n-% 

 p "p p <V : u p 



which take the place of I P) m p , n p when — p p is substituted for p p . 



Similarly, we will express r 1 (p p ) by £ ; (p p ). 



If equation (15) be now multiplied by l v and the analogous 



expressions modified by changing £ and / into rj and m or into f 



