[150] PROFESSOR STOKES, ON THE 



positions of the flame and of the eye, both in this experiment and in that with a plane mirror, 

 the rings or bands seen in the two cases may be perceived, independently of any theory, to 

 be evidently of the same nature. It is unnecessary here to describe at length the various 

 appearances presented, since they are noticed in the body of the paper, in connexion with the 

 theory. 



The first section contains the theory of the rings formed in Newton's manner. The 

 investigation, though differing a little in the mode in which it is conducted, is the same in 

 principle as that given by Sir John Herschel, but is somewhat more general, inasmuch as the 

 curvatures of the two surfaces are supposed to be any whatsoever, and the luminous point is 

 not supposed to be situated in the axis. The distance, too, of this point from the axis is at 

 first supposed to be arbitrary, in order to investigate under what circumstances the rings can 

 be formed most dictinctly on a screen. The second section contains the theory of the bands 

 and rings formed by a plane mirror. The expression for the retardation is deduced as a 

 particular case from the formula investigated in the first section ; but on account of the interest 

 which attaches itself to these bands, and the simplicity of their theory, a separate investigation 

 is likewise given. The next two sections are devoted to cases of more generality, and on the 

 whole less interest : still, a few results of some interest are obtained. The last two sections 

 contain a closer examination of the precise mode in which the phenomena are produced. 



Although the present paper is a little long, the reader must not suppose that the theory 

 of the rings and bands is anything but simple. The length arises partly from the detail in 

 which the subject has been considered, partly from the generality of some of the investigations, 

 partly from the description of experiments which accompanies the theoretical investigations. 



Section I. 



Bings thrown on a screen by a concave mirror consisting of a lens dimmed at the first 

 surface, and quicksilvered at the back. Condition of distinctness when the rings are thrown 

 on a screen, or of fixity when they are viewed in air. Investigation of the phenomena 

 observed when the luminous point is moved in a direction perpendicular to the axis of the 

 mirror. 



1. Let a luminous point L be situated either in or not far out of the axis of a mirror 

 such as that just described; and let it be required to investigate the illumination, at the point 

 M of a screen, due to two streams of light, of which one is scattered at the first surface, and 

 then regularly reflected and refracted, and the other is regularly refracted and reflected, and 

 then scattered in coming out, the point M being supposed to be situated not far out of the 

 axis. Let the mirror be referred to the rectangular axes of x, y, z, the axis of z being the 

 axis of the mirror, and the origin being situated in the first or dimmed surface. Let r be the 

 radius of the first surface, s that of the second, t the thickness of the glass, m its index of 

 refraction ; and suppose r and s positive when the concavities of both surfaces are turned in 



