618 Transactions of the Society. 



and F will be out of phase, that from E lagging behind, that from 

 F leading by a certain amount expressed as an angle, depending on 

 the distance between E or F and A compared with the distance 

 between A and B, and on the order of the spectrum. Let this 

 difference of phase be ft, then the light emanating from E will 

 cause the disturbance — 



(2) x K = c . sin (a - ft), 



the light from F the disturbance — 



(3) x = c . sin (a -f- ft). 



Any pair of points equidistant from the centre of the slit will 

 produce a pair of disturbances like (2) and (3) ; and, as the order 

 in which the elementary disturbances are added together is im- 

 material, we will combine them in such pairs. 

 Now, solving the sines in (2) and (3) we get — 



x E = c sin a cos ft — c cos a sin ft 



x t =c sin a cos ft + c cos a sin ft, 



and adding the two together we get — 



II. x a + os, = 2 c cos ft . sin a. 



ft and cos ft being constants for any one pair of points, 2 c cos ft 

 is obviously the amplitude of the combined vibrations, and we see, 

 firstly, that the intensity resulting from such pairs away from the 

 centre is less than that from a similar area in the centre of the 

 slit in proportion of cos ft : 1 ; secondly, that the combined phase 

 is the same as that of the central wavelet as long as cos ft remains 

 positive ; and thirdly, that the phase is reversed when cos ft be- 

 comes negative, which happens when ft becomes greater than 90° 

 but less than 270°. To render this quite clear, and also to facilitate 

 the study of these phase relations, I will tabulate the amplitudes 

 of successive pairsffrom the value of 2 c cos ft according to II.,. 

 taking pairs covering 15° difference of phase, the first pair to cover 

 the centre of the slit ; and, for simplicity's sake, I will put c = £, 

 its value being quite immaterial to this inquiry. 



In the third column I have added up the values contained in 

 the second, thus carrying out roughly what mathematical integra- 

 tion does accurately, and determining approximately the total 

 amplitude caused by a broad slit extending over all the successive 

 pairs it embraces, but the result is sufficient for this discussion. 



It shows that, given an arriving wave of a certain intensity, 

 and assuming slits the width of which increase from indefinite 

 smallness, then the intensity of any diffraction spectrum will 



