OF NEWTONS RINGS BEYOND THE CRITICAL ANGLE. 645 



difficulty. In this case, whether the angle of incidence be greater or less than the critical ano-le, 



V denotes a displacement, or else a velocity, perpendicular to the plane of incidence. When the 

 vibrations are in the plane of incidence, and the angle of incidence is less than the critical angle, 



V denotes a displacement or velocity in the direction of a line lying in the plane xss, and inclined 



at angles ir - i', - I - - i'\ to the axes of x, z, i being the angle of refraction. But when the 



angle of incidence exceeds the critical angle there is no such thing as an angle of refraction, and 

 the preceding interpretation fails. Instead therefore of considering the whole vibration V', consider 

 its resolved parts F/, F/ in the direction of the axes of .r, z. Then when the angle of incidence is less 

 than the critical angle, we have 



rj - - «' F' = - cos r . F' ; IV = l'V' = sin i' . F', 



F' being given by {A), and being reckoned positive in that direction which makes an acute angle 

 with the positive part of the axis of z. When the angle of incidence exceeds the critical ano-le, we 



ir 



must first replace the coefficient of F' in F/, namely - n, by v'e^^~^, and then, retaining v 

 for the coefficient, add — to the phase, according to what was explained in the preceding article. 



Hence, when the vibrations take place in the plane of incidence, and the angle of incidence 

 exceeds the critical angle, F' in (2?) must be interpreted to mean an expression from which the 

 vibrations in the directions of x, z may be obtained by multiplying by i/', /', respectively, and 



increasing the phase in the former case by — . Consequently, so far as depends on the third 



of equations (D), the particles of ether in the second medium describe small ellipses lying in 

 the plane of incidence, the semi-axes of the ellipses being in the directions of a;, z, and being pro- 

 portional to (/', I', and the direction of revolution being the same as that in which the incident ray 

 would have to revolve in order to diminish the angle of incidence. 



Although the elliptic paths of the particles lie in the plane of incidence, tiiat does not prevent 

 the superficial vibration just considered from being of the nature of transversal vibrations. For it 

 is easy to see that the equation 



d v; d V.' 



— - + =0 



dx dz 



is satisfied ; and this equation expresses the condition that there is no change of density, which is the 

 distinguishing characteristic of transversal vibrations. 



5. When the vibrations of the incident light take place in the plane of incidence, it appears 

 from investigation that the equations of condition relative to the surface of separation of the two 

 media cannot be satisfied by means of a system of incident, reflected, and refracted waves, in which 

 the vibrations are transversal. If the media be capable of transmitting normal vibrations with 

 velocities comparable with those of transversal vibrations, there will be produced, in addition to 

 the waves already mentioned, a series of reflected and a series of refracted waves in which the 

 vibrations are normal, provided the angle of incidence be less tiian either of tlie two critical angles 

 corresponding to the reflected and refracted normal vibrations respectively. It lias been shewn 

 however by Green, in a most satisfactory manner, that it is necessary to suppose the velocities of 

 propagation of normal vibrations to be incoin[)arably greater tiian those of transversal vibrations, 

 which comes to the same thing as regarding tiie ether as sensibly incompressible; so that the two 

 critical angles mentioned above must be considered evanescent*. Cmiseciuently the reflected and 



• Camfjri'tt/f: I'liil'imitUual TrumtactiofiH, Vol. vil. p. 



Vol.. VIII. Part V. lO 



