M. Lorenz on the Theory of Light, 215 



others derived from it by the exchange of 1 for 2 and for 3. Since 

 this sum is not altered by exchanging u for v } or I for m, we con- 

 clude, as before, that we shall have a 1)2 = a 2} i. In the same way 

 we have a h 3 = #3, i and a 2} 3 = a 3f 2 . It is easy to show further that 

 the odd powers of oc p disappear in the coefficients a. 

 The coefficient of |' is 



S l 2 3 \u x («j ^2, i + u 3 $z, i ) — uj (u 2 3-2,1 + % 3 3j r)] sin A , 



The sum is, however, equal to nought, whence b x . x = ; and simi- 

 larly, 5 2 ,2 = 0, and £3,3 = 0. 

 Lastly, the coefficient of rjj is 



e, e 2 63 



S {3 [»i(«2 %, i +^3 s 3 ~; 1) — t»T (^2^2,7+^3^3,7)] sin A. 



If n is here exchanged for v, or Z for m, we get the same value with 

 the contrary sigD, thus b 1)2 = — b 2 ,i' } and similarly, Z> 1>3 =— 5 3>1 , 

 b 2)3 =—b 3)2 . It is easy to see also that the coefficients b will 

 contain only odd powers of a p . 



From the equations (C) and the relations that have been found 

 between the coefficients, it will be at once obvious to any one 

 acquainted with the theory of polarized light, that this theory is 

 completely contained therein. I nevertheless take the liberty to 

 indicate this by a few lines a little more distinctly. 



We can, as before, choose the direction of the coordinates so 

 as to satisfy the equations 



a 1)2 =0, a 1>3 = 0, 02,3=0. 



We will further put 



_o 2 _xx 2 _n 2 



#1,1 — ~2~j #2,2— ~TT) #3,3— —f[i 



b^=o.% b 3)1 =n\ 6 1)2 =ny. 



The values on the right of the equations (C) are 



qz _ 



^(fo) = 7 tfo- < u ?o + v Vo + w &)] , 

 &c. 



These values having been transferred to the left side, we may 

 write 



1 I-u 2 _ 1 l-v <2 _ 1 l-w 2 _ 



,,2 2 — C l,l) m c 2 — C 2,2J ~2" 2 "~ C 3, 3 j 



VW_ _ UW _ _ UV __ 



—a — ^2,3 — £3,2; "~2* — C l,3 — C3, ly "To" — C l,2 — <?2, 1* 



6 S S 



To the three equations thus formed correspond three other ana- 



