of the Interference of Homogeneous Light. 87 



We have, as will be easily seen from what has preceded, 

 the difference of the thickness of glass which has been passed 

 through by the two rays, equal to 



the distance (on + sq) = h tan $' + h' tan \J/, 



and the differences in air equal to 



the distance (m p + rt) = \l h tan $' + /x, h 1 tan ty. 



Now these rays diverging from another point a' after the 

 refractions, their relative positions will depend on the velocity 

 with which they have traversed the glass of the prism ; and 

 by an analogous procedure to that which we used in the last 

 article, and considering ? x now to be variable, we find the dif- 

 ferential of the radius vector r to be 



/ztan<p' + #tan\[/ 



dr = ft (h tan <$>' + ti tan \[/) cv> 



m 



or. 



dr — (h tan $f + h ! tan 4/') (<p cv> V 



We may now apply the theoretical values for m ; and ac- 

 cording to the undulatory theory where m = — , we have 



dr = (7z tan 4>' + #' tan \J/)(V p J = (Atan^ f +^tan 4/)(/x— ft) ? 



and fd r = r = constant ; ^ 



which is independent of the values of <p f and \!/, and we recog- 

 nize the polar equation of the circle referred to the centre. 



As this equation has been arrived at rigorously, without 

 any approximate considerations, and as we cannot integrate 

 in the same rigorous manner for the Newtonian hypothesis, 

 I shall proceed in the first place to the examination where 

 interference should arise according to the undulatory theory. 

 Referring therefore to fig. 4, and taking the positions of 

 the secondary images of the luminous point, as we found them, 

 in a! and b\ and the simultaneous positions of the undulations 

 on the axes of the pencils as we found them to be, on the same 

 perpendicular sf, we have, making the point m' the origin of 

 the rectangular coordinates, m'g= a' = m'b\ b'g = 2 a', which 

 will always bear a determinate ratio to 2 a depending on the 

 incidence, and at the angle of minimum deviation it will be 

 that of equality or 2« f = 2 a, in which case we now take it: 

 and as we found before that we may calculate the distance g a' 

 in terms of 2 a, we will call this distance m a. Then m' being 

 the origin, and the lines m f y 9 m'x the axes of the coordinates, 



