88 M. Lorenz on the Theory of Light. 



The two compound rays, reflected and refracted, represent the 

 actual movement of the light in the substance. If the vibrations 

 take place perpendicularly to the plane of incidence, the direction 

 of vibration is the same in both rays, and the actual amplitude 

 is then the sum of the two expressions (11) and (13). If, on 

 the other hand, the vibrations take place in the plane of incidence, 

 they make different angles with the axes in the two rays. They 

 are, as already stated, perpendicular to the ray; and if the posi- 

 tive direction is taken as to the left of the observer when turned 

 towards the ray, the vibrations of the incident ray make in the 

 layer (a) an angle of 90° — oc with the axis of x, and an angle of 

 180° — a with the axis of z ; while in the reflected ray they make 

 an angle of 90° — a. with the axis of x } and of a with the axis 

 of z. Hence, if the components of the actual excursion in the 

 direction of these two axes be denoted by f and f, we have 



|?=sin« / ,P e -<V-(u-2), • • • (14) 



?=-co S « /) P e -^ r "'(u+2)- • • ( 15 ) 



The function U is determined by (12), or by the differential 

 equation deduced from this value, 



** 3 y-^.^ =U (16) 



If we now put 



du 2 du du 



e- ** 



(»-£)=•. r*«(u + 8H ■ <»> 



7C\ 



we have, by writing dS in place of -z-du, 



de u s ,—-= , , de~ u s' ,— = 



whence 



L 



dh 



(^§) +rt =0, i(^')+W=0. (18) 



We will further substitute other expressions for sin a and cos a 

 in (14) and (15). Let it be remembered that by (5) and (6) the 

 amplitude for the simple transmitted ray was determined by terms 

 of the form 



In the element of time dt the motion has been propagated 

 through the small space cos a dx or sin a dz with the velocity <d, 



