PROCEEDINGS OF THE SOCIETY. 735 



corresponding values x v ■x_ 1 , j3 x for the first pair, and so on to 

 x n i x -m fin ^ or the ^ as ^ P an ' '■> then we shall have n equations of the 

 form of II., differing only in the value of the phase-difference (3 which 

 grows proportionately to the distance of the strips from the centre of 

 the slit. The sum to be formed is therefore composed of the terms 



•*'i + £_i = 2 c cos /?! sin a 

 x 2 4- x_ 2 =2 c cos /3 2 sin a 



x n + x - n = 2 c cos (3 n sin a. 



All these terms have the common factor 2 c sin a, and the sum, or total 

 disturbance, is therefore 



m = n 



III. 2 (x m + x_ m ) = 2 c 2 cos j3 m . sin a. 



m =1 m=l 



The successive values of the /3 being independent of time, and there- 

 fore constant for a spectrum of given order from a given grating, the 

 sum on the right is also a constant, and combines with 2 c to form the 

 amplitude of the integral wave ; and this amplitude has been tabulated 

 in Table I., column 3, for the very coarse interval there adopted. 



The step from the sum in III. to the integral, i.e. the correspond- 

 ing sum taken with pairs of indefinitely narrow strips, is now exceed- 

 ingly simple. We introduce a new constant, C defined by c = G d (3, 

 <•? /3 representing the infinitesimal difference of phase between the extreme 

 edges of any one of the elementary strips, and we can then at once write 

 down the integral expressing accurately the total effect of the light, 

 from the whole slit ; putting the limiting value of the angle /? for the 

 extreme edges of the slit as [3$ and the integral disturbance as K, we find 



cos (3 d (3 x sin a = 2 G sin /fy X sin a ; 



= 



and, of course, (3^ being again a constant, we identify 2 G sin (3$ as the 

 amplitude of the resulting wave, whilst sin a again expresses the un- 

 dulatory nature of light, inasmuch as it goes through its complete 

 cycle of positive and negative values for every increase of a by 2 v. 

 But the amplitude is the quantity of interest, and this is therefore 

 tabulated in column 4 of the table, to show that it agrees closely with 

 the result of the rough mechanical integration tabulated in column 3. 



The important conclusions to be drawn from this table are contained 

 in the original paper, and need not be again referred to. 



It will now be clear that where Mr. Gordon errs is in claiming that 

 a is also the value of the phase difference for the edges of the slit — which 

 latter / have consistently designated by (3. He gives no justification 

 for this, and could not ; for a is an angle ever growing with time at 

 a tremendously quick rate and to enormous values, whilst the j3 are 

 comparatively small differences of phase which are independent of time 

 and limited to a moderate number of wave-lengths. 



It is, therefore, no wonder that Mr. Gordon's "solution" of the 

 integral, i.e. 3C = 2 C sin 2 a, is of a form which — assuming that a retains 



