DISPERSION IN ARTIFICIAL DOUBLE REFRACTION. 267 



The relative retardation after traversing the first slab for any ray which passes 

 through P, at a small angle to the horizontal = 2a, . C| (3Miyi/4a 1 /> 1 J ). 



For, although the stress is not uniform along the path of the ray, the mean stress 

 along the ray = stress at the middle point, since the stress varies linearly as the 

 distance from the neutral axis. Also the length of the ray differs from the breadth 

 of the slab only by quantities of the second order. Hence the result above. 



Accordingly the total relative retardation, after passing through the two slabs, is 



given by 



...,., (4). 



Substituting for y t from (3) into (4), 



R = (3M./26, 3 ) (^.-(SM^fe,') C^+(3M,/2V)C s [( 2l +y I ) (i + -)-crfc] (5). 

 Now, if R is to be independent of y,, we must have 



= ....... (6). 



This condition will not of course be accurately fulfilled for all colours at the same 

 time ; in the first place, because C, and C a do not in general follow the same law of 

 variation for the two slabs ; in the second place, because <r contains ft, and therefore 

 involves the wave-length. 



It is easy to see that the latter cause of error is quite negligible. For if d be 

 large compared with a, or a,, which was the case in all the experiments, the error 

 introduced in o- by a variation 8/x in the refractive index (taking /i, = /*,, which is 

 practically true) is approximately 



Now, a, + a, = 3 centims. in the experiments to be described ; d = 250 centims. 

 about (or larger), 8p. = O'Ol between the C and F lines of the spectrum which 

 represent fairly well the extreme range of the observations. 



Hence, ft being about 1 '5, 



(oi + o^S/i/pU = -01/187-5 < -00006. 



Accordingly the error introduced by this cause would correspond to an error in C, 

 of less than 6 in a hundred thousand, an error which is absolutely negligible, since 

 the errors of ordinary observations in the method to be described amount to ffaj or 

 T&o of C. (See ' Camb. Phil. Soc. Proc.,' vol. XII., p. 58.) 



The di fii-rent variation of d and C, with the wave-length would be far more 

 important This, however, need not be considered, for the two slabs N and F are 

 taken from the same cast, so far as possible, so that C, and C, should be identical. 

 In some cases it was found that C, and C 2 differed ; but, at the same time, the 



2 M 2 



