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SCIENCE PROGRESS 



mately in the order of their intensities, some spots being much 

 darker than others, the following list is the result : 



Table I 



h, 



2 



I 



5 



i 



7 



3 

 i 



2 



3 

 5 

 7 

 7 

 7 

 5 



h 3 



i 



i 



i 



i 



3 



3 



2 



Each set of numbers in this table corresponds of course to four or eight spots 

 in the pattern, according as hi and h z are equal or unequal. 



I.e. 5, 3, i represents ± 5 ± 3 1 



± 3 ± 5 1 



3, 3, 1 represents ± 3 ± 3 1 



the pattern being of fourfold symmetry. 



Though all the numbers are simple ones, it is not at once 

 obvious that they belong to any system. Why, for instance, should 

 there be spots in the photograph for which hi, h 2 , h 3 have values 

 1, 3, 1 ; 1, 4, 1 ; 1, 5, 1 ; 1, 7, 1 ; and no spot corresponding to 

 1,6, 1 ? Also there are sets 2, 2, 1 ; 3, 3, 1 ; 5, 5, 1 ; but no set 

 4, 4, 1. There are many similar gaps in the series. Again the 

 most intense spots are not given by the simplest value of hi, h 2 , h 3 

 in this list, as would seem natural, these spots being analogous 

 to the spectra of low orders in the case of a diffraction grating, 

 which are generally the brightest. A theory of the effect must 

 attempt to explain these anomalies. 



On considering equations (1), which for convenience are here 



repeated : 



a. a = hiX 



a/3 rrrhaA 



a(i -y) = h 3 X 



it is clear that a knowledge of the numbers h 1( h 2 , h 3 to be assigned 

 to any spot determines the wave length of the radiation which 

 has at that point an interference maximum, as well as the direction 

 cosines of the pencil which forms it. There are three equations 

 to be satisfied. The values of afty represent only two variables 

 since they must obey the equation : 



a 1 + /3 2 + y 3 = I 



and therefore the value of A. must also be adjusted to satisfy the 

 equations. It is here that the action of a " three dimensional " 

 grating differs from that of a " one dimensional " grating like an 



