174 



First, let the total reflexion take place at the common 

 surface of two ordinary media, as between glass and air, and 

 let it be proposed to determine the incident and reflected 

 vibrations, when the refracted vibration is known. It is to 

 be observed, that the refracted vibration (which is in general 

 elliptical) cannot be arbitrarily assumed ; for, as may be 

 inferred from what has been already stated (Proceedings of 

 the Academy, vol. ii. p. 102), it must be always similar to the 

 section of a certain cylinder, the sides of which are perpen- 

 dicular to the plane of incidence, and the base of which is 

 an ellipse lying in that plane and having its major axis per- 

 pendicular to the reflecting surface, the ratio of the major to 

 the minor axis being that of unity to the constant r. The 

 value of r, as determined by the general rule in p. 101, is 



r = \/l ~ -^■, 



where i is the angle of incidence, and n the index of refrac- 

 tion out of the rarer into the denser medium. The ellipse is 

 greatest for a particle at the common surface of the media ; 

 and for a particle situated in the rarer medium, at the 

 distance ;? from that surface, its linear dimensions are pro- 



2nrz 



portional to the quantity e~~^; so that for a very small value 

 of s the refracted vibration becomes insensible. 



Now, taking any plane section of the aforesaid cylinder 

 to represent the refracted vibration for a particle situated at 

 the common surface of the two media, let op and oq be the 

 semiaxes of the section, and let them be drawn, with their 

 proper lengths and directions, from the point of incidence o ; 

 through which point also let two planes be drawn to repre- 

 sent the incident and reflected waves. Then conceive a 

 plane passing through the semiaxis op, and intersecting the 

 two wave-planes, to revolve until it comes into the position 

 where the semiaxis makes equal angles with the two inter- 

 sections ; and in this position let the intersections be made 

 the sides of a parallelogram, of which the semiaxis op is the 



