On FresnePs Formulas for Reflected and Refracted Light. 267 



dence, we have the obvious geometrical construction of the tri- 

 angle formed by the known directions of the rays, and conse- 

 quently of the amplitudes at right angles to them, and in the 

 same plane (first paper, § 23) . In this triangle, the angles being 

 (e— r), (2-fr), and {7r—2i), the ratios of the sides will be (as 

 before) 



I _ 



sm 2i 



h! sin {i — r) h 



h ^m{i-\-rY A sin(i + r)* 



5. On MaccullagVs theory oi equal densities, it will be remem- 

 bered that this construction represents the mechanical equiva- 

 lence, and the sides thus give the values of the amplitudes. 



The close agreement of these values with those for the ampli- 

 tudes in case (a) deduced on the hypothesis of increased density, 

 — to which no such construction can apply, — has been already 

 remarked (first paper, §§ 25, 41). 



But these values cannot be those of the amplitudes in case (/3) 

 on the hypothesis of increased density, since experimental results 

 essentially require a different relation in this case from that in 

 case (a). 



6. From the triangle, however, we have this obvious and 

 simple relation of equivalence, 



h^ = h cos {i—r) + A' cos [i + r) ; 



or expressing these components parallel to h^ by [h) [h!), we have 



But in this construction, it must be borne in mind that if we 

 assume the side h^ as the value of the refracted amplitude, it is 

 represented geometrically on the same scale with the other sides 

 by which we measure the components ; and for its actual value 

 it is necessary to take into account its diminution in the more 

 retarding medium in the ratio of the refractive index ; or if {h^) 



