On the Problem of Total Reflexion. 219 



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

 particle situated in the rarer medium, at the distance z from that 

 surface, its linear dimensions are proportional to the quantity 



_2jrns 



e A ; so that for a very small value of z the refracted vibra- 

 tion 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 semi- 

 axes of the section, and let them be drawn, with their proper 

 lengths and directions, from the point of incidence ; through 

 which point also let two planes be drawn to represent the inci- 

 dent 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 intersections ; and in 

 this position let the intersections be made the sides of a parallel- 

 ogram, of which the semiaxis OP is the diagonal. Let OA 

 and OA', which are of course equal in length, denote these two 

 sides. Make a similar construction for the other semiaxis OQ, 

 and let OB, OB', which are also equal, denote the two sides of 

 the corresponding parallelogram. Then will the incident vi- 

 bration be represented by the ellipse of which OA and OB are 

 conjugate semidiameters, and the reflected vibration by the 

 ellipse of which OA' and OB' are conjugate semidiameters. 

 And the correspondence of phase in describing the three ellipses 

 will be such that the points A, A, P will be simultaneous posi- 

 tions, as also the points B, B', Q. 



The same construction precisely will answer for the case of 

 total reflexion at the surface of a uniaxal crystal, which is 

 covered with a fluid of greater refractive power than itself. It 

 is to be applied successively to the ordinary and extraordinary 

 refracted vibrations, and we thus get the uniradial incident and 

 reflected vibrations, or rather the ellipses which are similar to 

 them. And as any incident vibration may be resolved into two 

 which shall be similar to the uniradial ones, we can find the re- 



