TiY A. L. McAlLAY. r..Sc., B.A.. Pli.D. 97 



The essential difference between the two cases lies in the 

 fact that en further increasine: '^ in the case of the j^rating, 

 a point is reacherl where the resultant disturbance from one 

 clement diflers frcaii that from the one next it by exactly one 

 wave length, i.e., the disturbances over q (Figure III.) are all 

 in phase again. This is evidently where 1 a'lu ^ \ (see 

 Figure VI., section 6), and results in the first order spectrum. 

 This can never occur for the slit, because the elements are in- 

 finitely close together, and 1 = 0, 



Refer to Figure IV., section 4, and consider the illumina- 

 tion at a point Q, as represented by the distance between the 

 ends of the jointed rod or chain. As Q moves away from P, 

 the chain passes through the stages 1, 2, 3, 4, and 5, and then 

 continues ceiling up on itself, the successive maxima becom- 

 ing smaller and smaller. In the case of the grating, there 

 comes a point where one .section is turned back compl'^tely 

 ever the cne next it, and a further rotation of the sections 

 begins to uncoil the chain once more, the uncoiling continuing 

 till one section has turned through 360 degrees relatively to 

 the one next it, and the chain is again stretched straight out. It 

 is here that the first order spectrum is found. In the case of 

 the slit the sections being infinitesimal, the chain continues 

 to coil up for ever, degenerating to a point when Q reaches 

 infinity. 



The above shows the reason for the existence of the 

 spectra and the secondary maxima near the central line, 

 but a little further consideration is needed to extend the dis- 

 cussion to the region in the neighbourhood of the spectra and 

 to find the position of the first minimum associated with a 

 rpectral line. 



Referring to Figure VII., p is a wave front of the beam of 

 rays which goes to form the central maximum of a spectrum, 

 and q is the plane normal to the bundle which is united at the 

 first minimum flaniiing the spectral line. Then the dis- 

 turbances at C on both p and q are in the same phase, and the 

 disturbances at C and B are in the same phase. Therefore, 

 the phase difference between extreme rays of q is the same 

 as the phase difference between B and D which is evidently 



"- [K sin (0 + <\ 0) - \: r-iu 0]. i\ is Very small. Identifying 



C'PH >\0 witli 1 mikI (-ill <!/' wiili i\(), this expression becomes 



^ ]•: CO.* e (1 0. F'rom section 4, Figure IV., the first mini- 



mum occurs when the phase difference between extreme rays 



of 11 i» 2 IT, i.e., \\\<''n ~^ Ei-'^s II d e-^-l IT or when A O = — 



K co« tf. 



