86 M. Lorenz on the Theory of Light. 



As the ray penetrates further into the substance, it undergoes 

 continually a partial reflexion, while in the transmitted portion 

 both A and the exponent vary. The expression for the excur- 

 sion in a layer which we may call the layer (a), wherein the 

 angle of incidence is a, takes consequently the form 



pVe -s s/~i } . (6) 



in which p and 8 (the ' ' retardation of phase ") are functions of x. 



In the following layer a becomes a + da, and p becomes p + dp, 



and, by (3) and (4), whatever the direction of vibration may be, 



L ^"~~"\sin2a tana/ 



From this equation we get by integration 



P = V ta^ViiW' W 



where a denotes the angle of incidence for p = l, or for the point 

 where the ray enters the substance. 



For the ray reflected at this same layer (a -f- da), p becomes 



Xp-* — —-when the vibrations are in the plane of incidence, and 

 r tan 2a 



da. 

 + p . - when they are perpendicular thereto. Let both these 

 — r sm2a J 



values be denoted by pdu, where, in the first case, 



w= + J log sin 2a, (8) 



and in the second 



w=±^logtana (9) 



For the various layers (a x ), (a 2 ), &c, we will use the same 

 symbols p, 8, and u, distinguished by the indices 1, 2, . . . &c, 

 respectively, and we will denote the corresponding functions for 

 the layers (a) and (b), which we will suppose to be the last layer 

 of the substance, by p a , S a , u a , and p b , o b , u b . 



The amplitude of vibration for the ray reflected at the layer 

 (a 2 ) is accordingly 



p^e- s ^~ l du v 



"When this ray, whose retardation of phase i3 here S 2 , passes 

 into a subsequent layer (a 2 ), its phase is further retarded by 

 \— Sj ; the corresponding amplitude is therefore now 



At the surface of the next following layer the ray is again par- 

 tially reflected, and the amplitude for the reflected portion is 



