Prof. Lorenz on the Theory of Light. 417 



the plane of the wave, we have 



sin a— ± * 8 ' * b * , 



whence follows 



t a n q= + %. + ™l. + *Z. 



We get, further, from the above values of rf and f, 

 k(n V '-m?)=l(m Vs + nQ-(m* + n*)l. 

 By eliminating f 8 from these two equations, after having ex- 

 changed the denominator of the first for k\/% n + ?/ 2 + J' 2 , we have 



m V + n£= \ 2 [/(nV-m?') ±K + ^ 2 ) tan g*/? 5 + v'' 2 + S' 2 ] J 



and this expression must therefore have the same value on both 

 sides of the limiting surface. If, adopting Neumann's notation, 

 we call the angle which the plane of the wave and the plane of 

 refraction (yz) make with each other <£, and the angle which the 

 new resultant makes with the line of intersection of the plane of 

 the wave and the plane of refraction yfr, we have 



1 a 



— = COS 0, 



nrj -m%=z± cos ^v/V + m 2 \/%' 2 + V 2 + ? /2 . 



By introducing into it these magnitudes, the last limiting equa- 

 tion becomes 



[V?' 2 + V 2 + ? ,2 (cos 4> sin <£ cos >f± sin 2 tan g)]*^=0. 



This condition agrees exactly with Neumann's hypothesis, if 

 only we take the double sign arbitrarily, as Neumann did. 

 Strictly speaking, Neumann started from the assumption that 

 the intensity of the incident light is equal to the intensity 

 of all the refracted and reflected light; he, however, refers 

 this assumption back to the previous one. Conversely, there- 

 fore, we may conclude that our four limiting equations include 

 the principle of the maintenance of the intensity, and hence that 

 our intensity equation, by which this condition must be fulfilled, 

 is rightly chosen. 



Now that we have thus deduced Neumann's hypotheses from 

 our own, the problem we had proposed to ourselves is fully 

 solved ; for it follows from the results obtained that the same 

 theory of reflexion and refraction, complete and accordant with 

 experiment, which Neumann has developed, can likewise be 



