OF NEWTON'S RINGS BEYOND THE CRITICAL ANGLE. 647 



only be necessary to observe to take ± ,/ \/ - i, or ± v'\/- 1 with the negative sign. If we had 

 chosen to employ the expressions {B) with the opposite sign in the index, which would have done 

 equally well, it would then have been necessary to take the positive sign. 



8. We are now prepared to enter on the regular calculation of the intensity of the central 

 spot ; but before doing so it will be proper to consider how far we are justified in omitting the 

 consideration of the superficial undulations which, when the vibrations are in the plane of incidence, 

 are the representatives of normal vibrations. These undulations may conveniently be called normal 

 superficial undulations, to distinguish them from the superficial undulations expressed by the third 

 of equations (D), or the first and second of equations (F), which may be called transversal. The 

 former name however might, without warning, be calculated to carry a false impression ; for the 

 undulations spoken of are not propagated by way of condensation and rarefaction ; the disturbance 

 is in fact precisely the same as that which exists near the surface of deep water when a series of 

 oscillatory waves is propagated along it, although the cause of the propagation is extremely 

 difl'erent in the two cases. 



Now in the ordinary theory of Newton's Rings, no account is taken of the normal superficial 

 undulations which may be supposed to exist ; and the result so obtained from theory agrees very well 

 with observation. When the angle of incidence passes through the critical angle, although a materikl 

 change takes place in the nature of the refracted transversal undulation, no such change takes place 

 in the case of the normal superficial undulations: the critical angle is in fact nothing particular as 

 regards these undulations. Consequently, we should expect the result obtained from theory when 

 the normal superficial undulations are left out of consideration to agree as well with experiment 

 beyond the critical angle as within it. 



9. It is however one thing to show why we are justified in expecting a near accordance between 

 the simplified theory and experiment, beyond the critical angle, in consequence of the observed 

 accordance within that angle; it is another thing to show why a near accordance ought to be expected 

 both in the one case and in the other. The following considerations will show that the effect of the 

 normal superficial undulations on the observed phenomena is most probably very slight. 



At the point of contact of the first and third media, the reflection and refraction will take place 

 as if the second medium were removed, so that the first and third were in contact throughout. Now 

 Fresnel's expressions satisfy the condition of giving the same intensity for the reflected and refracted 

 light whether we suppose the refraction to take place directly out of the first medium into the 

 third, or take into account the infinite number of reflections which take place when the second 

 medium is interposed, and then suppose the thickness of the interposed medium to vanish. Conse- 

 quently the expression we shall obtain for the intensity by neglecting the normal superficial undu- 

 lations will be strictly correct for the point of contact, Fresnel's expressions being supposed correct, 

 and of course will be sensibly correct for some distance round that point. Again, the expression for 

 the refracted normal superficial undulation will contain in the index of the exponential - k I z, in 



r — ^. 



place of - /c V P 1^ », which occurs in the expression for the refracted transversal superficial 



undulation ; and therefore the former kind of undulation will decrease much more rapidly, in receding 

 from the surface, than the latter, so that the efl'ect of the former will be insensible at a distance from 

 the point of contact at which the efl'ect of the latter is still important. If we combine these two 

 considerations, we can hardly suppose the effect of the normal superficial undulations at intermediate 

 points to be of any material importance. 



10. The phenomenon of Newton's Rings is the only one in which I see at present any chance 

 of rendering these undulations sensible in experiment: for the only way in which I can conceive 

 them to be rendered sensible is, by their again producing transversal vibrations ; and in consequence 

 of the rapid diminution of tlie disturbance on receding from the surface, liiis can only happen when 



losi 



