M. Lorenz on the Theory of Light. 207 



give constant values by addition or subtraction. We will not, 

 however, consider this case until the next section. 



Two developments for tj and f analogous to equation (2) may 

 be obtained by changiug the letter f for 77 or f. 



We now get from equations (1) and (2) by multiplication 



+ s[?(±/) J ,) + |? +2|l(± /: , p ±p ? )]c(±p P ) + ...(3) 



H 2 _ 



This value, together with the corresponding values of — ^ 77, 



O 2 - ■ . ® 



— ?, is to be introduced into the right-hand member of the dif- 

 ferential equations (A) ; then, by comparison of the coefficients 

 of C, we get from the first equation (A), 



h + Zftt±P P ) = ^ P + m ^ n ^h-m <l + m % + nl)-}. (4) 



Corresponding equations may be derived from the two other 

 equations (A), and may also be deduced from (4) by changing 

 f for 7} and I for m, or f for f and I for n. We will denote the 

 expressions on the right of these three equations by &3(f ), <w(?7 ), 

 and ©(§>). 



By comparison of the coefficients of C( + p p ), G{ + p P + p q ), &c, 

 we may also get the coefficients f (±p p ), ~%( + p P '}:Pq), v{+pp)> 

 &c. expressed, with any degree of approximation, as linear func- 

 tions of f , ij 0} £)• Equation (4) and the two analogous equa- 

 tions thus acquire the following form : — 



0i,i?o + ai,a% + «i,3?o = ®(!o)j ] 



«3, l£o + «3,2?70 +«3, 3?0 —«(?<>)• J 



These equations determine the non-periodic portion of the 

 components of the excursion — that is to say, the sensible por- 

 tion of the motions of light ; for the periodic motions neutralize 

 one another, inasmuch as they disappear by integration over a 

 small extent. 



It can now be shown that the following three relations exist 

 among the coefficients a in the equations (B) ; namely, 



01,2 = 02, If 01,3 = fl, 3, lj 02, 3 = 03, 2 ') • • • (5) 



and that, when a p denotes infinitely small quantities, they are 

 independent of/, m, and n; whereas in the contrary case they 



