208 M. Lorenz on the Theory of Light. 



can be developed according to powers of /, m, and n, so that they 

 only occur as factors in even numbers. 



I may be allowed to confine the demonstration to the case in 

 which € p represents small quantities. By comparison of the 

 coefficients of 0(p p ), we get then only 



fW + f-fo-S^ + ^+OIW 



- l pUpZ(pp)+ m pV{pp)+npZ(pp)]\, • • ( 6 ) 

 where 



h = l+ -*-; m p = m+ J ?-; n p — n+—' 

 p p p 



The equation is multiplied by l p , and then two analogous equa- 

 tions are obtained by substituting first 77 and m, then f and n, 

 for f and /. By adding the three equations thus formed, we get 



l pHpp)+™ P y (pp)+n P Upp) + "2 ( l P%o + m pVo + npto) = ' 

 This, with the foregoing equation, gives 



= 2[\lp- l p 1 )Zo- l P m pV«-lpnp¥o]> ' * ( 7 ) 



by means of which the sum %-£%(±p p ) in equation (4) admits 



of being expressed as a linear function of | , 77 , £ . The coeffi- 

 cient of 7J , for instance, becomes 



*( 



fA 2 («p±l*p)(t>p± m * P ) 



{a p ±l« P ? + (b p ±m« p Y + ( Cp ± s« p ) 2 - J«/ 



and is the coefficient denoted in the equations (B) by a lf 2 . From 

 this expression we can also get a % x by putting b for a and m for 

 /; but since this would not alter the expression, we have 



#1, 2 := #2, 1« 



The truth of the two other equations (5) may be demonstrated 

 in the same way. 



If the above expression, or the value of any other coefficients, 

 is developed according to powers of oc p) the odd powers of these 

 magnitudes will cancel each other in the sum ; so that /, m> n 

 also only occur as factors in even numbers. 



