Royal Irish Academy. 1S7 



Professor Lloyd exhibited a specimen of rock from Terre Adele. 



Professor MacCullagh communicated to the Academy a very simple 

 geometrical rule, which gives the solution of the problem of total re- 

 flexion, for ordinary media and for uniaxal crystals. 



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 re- 

 fracted vibration is known. It is to be observed, that the refracted 

 vibration (which is in general elliptical) cannot be arbitrarily as- 

 sumed ; for, as may be inferred from what has been already stated 

 (Proceedings of the Academy, vol. ii. p. 102. Phil. Mag. S.3. vol.xxi. 

 p. 232), it must be always similar to the section of a certain cylinder, 

 the sides of which are perpendicular to the plane of incidence, and 

 the base of which is an ellipse lying in that plane and having its 

 major axis perpendicular 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 given in the place just 

 referred to, is 



= vA 



where i is the angle of incidence, and n the index of refraction 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 z from that surface, its 



2<rrz 

 linear dimensions are proportional to the quantity e \~ ; so that 

 for a very small value of z the refracted vibration becomes in- 

 sensible. 



Now, taking any plane section of the aforesaid cylinder to repre- 

 sent the refracted vibration for a particle situated at the common 

 surface of the two media, let O P and OQ be the semiaxes of the 

 section, and let them be drawn, with their proper lengths and di- 

 rections, from the point of incidence O ; through which point also 

 let two planes be drawn to represent the incident and reflected 

 waves. Then conceive a plane passing through the semiaxis O P, 

 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 parallelogram, of which the semiaxis O P is the diagonal. 

 Let O A and O A', which are of course equal in length, denote these 

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

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

 corresponding parallelogram. Then will the incident vibration be 

 represented by the ellipse of which O A and O B are conjugate semi- 

 diameters, and the reflected vibration by the ellipse of which O A' 

 and O B' 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 positions, 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 



