6-J8 Mr. stokes, on the formation of the central spot 



there exists a second reflecting surface in close proximity with the first. It is not my intention to 

 pursue the subject further at present, but merely to do for angles of incidence greater than the 

 critical angle what has long ago been done for smaller angles, in which case light is refracted in the 

 ordinary way. Before quitting the subject however I would observe that, for the reasons already 

 mentioned, the near accordance of observation with the expression for the intensity obtained when 

 the normal superficial undulations are not taken into consideration cannot be regarded as any valid 

 argument against the existence of such undulations. 



11. Let Newton's Rings be formed between a prism and a lens, or a second prism, of the same 

 kind of glass. Suppose the incident light polarized, either in the plane of incidence, or in a plane 

 perpendicular to the plane of incidence. Let the coeflicient of vibration in the incident light be 

 taken for unity ; and, according to the notation employed in Airy's Tract, let the coefficient be mul- 

 tiplied by b for reflection and by c for refraction when light passes from glass into air, and by e for 

 reflecuon and / for refraction when light passes from air into glass. In the case contemplated 6, c, 

 e, f become imaginary, but tliat will be taken into account further on. Then the incident vibration 

 will be represented symbolically by 



according to the notation already employed ; and the reflected and refracted vibrations will be repre- 

 sented by 



Consider a point at which the distance of the pieces of glass is D ; and, as in the usual investi- 

 gation, regard the plate of air about that point as bounded by parallel planes. When the superficial 

 undulation represented by the last of the preceding expressions is incident on the second surface, the 

 coefficient of vibration will become cq, q being put for shortness in place of e"*"^; and the reflected 

 and refracted vibrations will be represented by 



cqee"'"'. /<»''-MV-^ 



z being now measured from the lower surface. It is evident that each time that the undulation 

 passes from one surface to the other the coefficient of vibration will be multiplied by q, while the 

 phase will remain the same. Taking account of the infinite series of reflections, we get for the sym- 

 bolical expression for the reflected vibration 



{6 + cefq'(l + e'q' + e'q' + ...)] e*("'-"-+"--iv^. 



Summing the geometric series, we get for the coefficient of the exponential 



b + ^-!— . 



1 - e'q' 



Now it follows from Fresnel's expressions that 



b = - e, cf = 1 - e**. 



These substitutions being made in the coeflicient, we get for the symbolical expression for the 

 reflected vibration 



(1 -9")0 j,|i,f-to + ,ijK/~ /-yj^ , 



1 -q'b^ ^ ''■ 



• I have proved these equations in a very simple manner, without any reference to Fresnel's formulsp, in a paper which will appear in 

 ihe next number of the Cambridge and Dublin Mathematical Journal. 



