M. Lorenz on the Theory of Light. 91 



By now differentiating equation (28) in relation to z and in rela- 

 tion to x, and by putting -£- = — g f and ^S- = — -^ £ , we have 



d 2 £ d 8 f 1 d*% 

 dz* dxdz ft) 2 aft 2 



• • • 



dH, d 2 f 1 d 2 f 





<fo?? dxdz ~~ ft) 2 !&*-' 





(30) 



(31) 



The calculation can be conducted in the same manner when, 

 in the expression for u } the lower sign is taken, that is, when we 

 take 



but in this case the results are no longer equally simple. For 



ds rw 



instance, -j- — \f — ^ — n 9 - enters as a factor into the magnitudes 



f and \ , and hence it becomes impossible to determine them 

 by differential equations of the second degree. In order for this 

 to be possible, we are obliged to read the upper sign in FresnePs 

 formulae as they are given above, and we will not further pursue 

 the consequences of the opposite assumption. 



If the vibrations of the ray are perpendicular to the plane of 

 incidence, we have by (9), in case the upper sign be alone taken, 



u=i log tan a=const. —J log —• 



The actual excursion 77 is the sum of the two expressions (11) and 

 (13), or 



If we put 



<oPv=V, (32) 



we get 



^ =Al — 7M eVa ~ m) ' m ' • • • • • ( 33 ) 



where A x is, as before, a constant factor. The first equation 

 (18) now gives 



d 2 v /d$Y- Z 

 or 



^ + ^"ft> 2 dr : > - - ' ^ 



