X-RAYS AND CRYSTALS 381 



(— -] wave lengths, the co-ordinates of B being , — ; this 



must be a whole number. In order that this may be so, the 

 numbers hi and h 3 must be both odd or both even. The same 

 condition must hold for h 2 and h 3 . This at once explains the 

 peculiarities of the list of numbers h h h 2 , h 3 . Taking in that 

 list all the cases in which h 3 is unity, by the above rule h x and 

 h 2 must be odd. It is now clear why in so many sets h x and 

 h 2 have odd values and are even in two cases only, i.e. 1, 4, 

 1 ; 2, 3, 1. These last must now be written 2, 8, 2 ; 4, 6, 2 and 

 are "comparatively complicated. 



If the numbers h 1( h 2 , h 3 in the table be reconsidered, assum- 

 ing this new arrangement of the "elements" of the crystal 

 grating, the reason of their selection becomes clear. Take the 

 sets of numbers which have h 3 = unity. In the first place, by 

 calculating the corresponding wave lengths, they can be shown 

 to consist of every possible set of numbers which correspond to 

 wave lengths greater than a limiting value X = '034a. In the 

 second place, sets corresponding to a wave length approaching 

 X = -o6a give the two very intense spots 1, 5, 1 and 5, 3, 1 which 

 form a marked inner square in the photograph ; when the 

 wave length corresponding to a spot is greater or less than this 

 value, the spot is more faint, until spots corresponding to the 

 limiting wave length X = "034a can hardly be seen. It is as if 

 the incident radiation had a continuous spectrum with a maxi- 

 mum intensity at the region X = -o6a. If now the sets of 

 numbers having h 3 = 2 be considered, exactly similar results are 

 obtained. There are two very intense spots which may be 

 designated as 4, 6, 2 and 2, 8, 2, which have wave lengths 

 X= 013a and X = 055a, which form the outer square. But in 

 addition, there are others, such as 2, 4, 2 ; o, 6, 2 ; 4, 4, 2, which 

 are considerably fainter and have wave lengths further from the 

 maximum in the spectrum. The only difference between this 

 set and the one which has h 3 = 1 is that all the spots are com- 

 paratively fainter and that the range of wave lengths represented 

 is much reduced, there being now both an upper and a lower 



limit to the values of -. 



a 



The same may be said of the sets of numbers which have 



h 3 = 3, there being still fewer of these ; finally, only one very 



weak spot is visible corresponding to parameters which have 



