M. Lorenz on the Theory of Light. 87 



Finally, the ray reaches the layer (a), after a further retardation of 

 phase amounting to S— o\, with the amplitude 



The sum of the amplitudes of the rays which have been twice 

 reflected at all the layers is then given by the double integral of 

 this expression, u 2 varying from u x to u bi and u x then varying 

 from u a to u. This sum is therefore 



_ Cu Pu b _ 



-pPe-^-M duA i^gWi-^V-i. . . ( 10 ) 



whereas, according to (6), the amplitude for the unreflected ray 

 was 



Now let 



pP c -*V-iu ...... (11) 



be the amplitude for the compound ray, made up of refracted 

 rays that chave been reflected 0, 2, 4 ... . times ; we shall then 

 be able to express U by the equation 



U=l- \ U du\ ^r«V^ t b . (12) 



where U 2 denotes the same function of u 2 that U does of u. 

 Introducing into the expression this value of TJ 2 , we get TJ 

 expressed by an infinite series. The accuracy of the expression 

 (11) is apparent, not only from this evolution of an infinite 

 series, the several terms of which give the amplitudes of the 

 rays that have been reflected 0, 2, 4, . . . times respectively, but 

 from its being possible to conceive this compound ray as again 

 twice reflected at every layer, without the expression for the 

 amplitude undergoing any modification whatever. 



Again, the amplitude for the compound reflected ray, which has 

 suffered 1, 3, 5, . . . reflexions, will be expressed by 



-^- WZT f- ••••••• M> 



By (12) this expression is equivalent to 





and this integral is precisely the sum of the amplitudes of the 

 compound refracted ray which has been once reflected to the 

 layer (a). • : ; ; 



