310 BELL SYSTEM TECHNICAL JOURNAL 



tive rc-direction and the line from dS to P, so that cos {n, r) = — cos 6. 

 Consequently : 



4x5o = \ dS\ ^ cos {nt — mr) (1 + cos 6) sin {nt — mr) 



(74) 



One is tempted to say that the quantity under the integral sign is 

 the contribution made by the element-of-wave-front dS to the value of 

 5 at P. This notion facilitates both thought and description, and I 

 will adopt it, but with a warning. The danger is that one may come 

 to think of an element-of-wave-front as an independent entity, capable 

 of existing by itself in the medium regardless of what other elements-of- 

 wave-front adjoin it or stand elsewhere. This is unpermissible, for 

 the same reason which makes the method that I am now expounding 

 an approximate and not a rigorous one. Were one of these elements 

 of wave-front alone in the medium, the function 5 would not conform 

 to the wave-equation. Therefore if we call the expression 



dsQ = z—dS 

 47r 



1 fH 



-^ cos (nt — mr) (1 + cos 6) sin (nt — mr) 



(75) 



the contribution of the element-of-wave-front dS, we must always re- 

 member that it cannot be isolated, but — like the donations to certain 

 endowments — is given only under the condition that other elements 

 also contribute.^ 



The contribution of dS, then, is made up of two terms, one varying 

 inversely as r^ and the other inversely as r/m. At great distances the 

 latter must increasingly outweigh the former; and "great distances" 

 in this context signify those which are much greater than 1/m — that is 

 to say, very many times as great as the wave-length of the light. 

 Now as the wave-lengths of most kinds of light are less than .001 mm., 

 a field-point where observations can actually be made must necessarily 

 be distant by many wave-lengths from the screen. Hence it is cus- 

 tomary to ignore the first term in the expression (75) and in its integral, 

 and write for the contribution of dS: 



1 ffl 



dso = — -i-dS— (1 -f cos d) sin {nt — mr). (76) 



47r r 



This expression is the approximate description of what in an earlier 



^ The reader may notice that whereas in dealing with a closed surface surrounding 

 the field-point the w-direction was defined as that of the "inward-pointing normal," 

 there is no way of discriminating between the two senses of the normal to an isolated 

 area-element. This causes an ambiguity in the sign of the contribution ; for reversing 

 the sense of the normal reverses the signs both of dUjdn and of cos (n, r). The 

 ambiguity is always, I think, physically trivial. 



