320 BELL SYSTEM TECHNICAL JOURNAL 



Sq = — \ dS\ (cos 6 + cos if) sin («/ — my sin (p — mr) , 



47r J I r J 



and there are corresponding changes in the values of the integrals C 

 and S which determine the ampHtude. To first approximation — that 

 is to say, when (p is not too great — the result is, that the diffraction 

 pattern of one star is like that of the other, but shifted sidewise. The 

 angular displacement between the centres of the two fringe-systems is 

 the same as the angular displacement between the two stars. The 

 question now arises: how far apart must the two fringe-centres be, 

 that the two families of rays may be securely told apart? 



Such a question of course cannot be definitely answered ; the answer 

 would depend upon the acumen and the experience of the observer. 

 The conventional response is, that the two systems of rings are surely 

 distinguishable if the centre of one lies upon the first dark circle of the 

 other. Now the angular radius of the first dark ring, i.e., the value of 

 /3 for which the Bessel function of (97) first vanishes, is 1.22 times the 

 ratio of the wave-length of the light to the diameter of the aperture; 

 for green light in the largest available refracting telescope this amounts 

 to about an eighth of a second of arc. This then is nearly the least 

 angular separation between two stars which are distinguishable; a 

 pair or a group much closer together would appear as one, not through 

 any avoidable defect of the telescope nor through any insufficiency of 

 the eyepiece but through the laws of propagation of light themselves, 

 working to prevent the formation of an image indefinitely sharp. 



For a rectangular aperture the integrals C and 5 are extremely easy 

 to evaluate. The diffraction-pattern is a criss-cross of dark lines, 

 intersecting at right angles and bounding rectangular areas of light, 

 similar in shape to the aperture but oriented at right angles to it. 

 If the rectangle is prolonged indefinitely and so becomes an infinitely 

 long slit, the diffraction-pattern becomes a sequence of parallel bands 

 separated by dark lines normal to the length of the slit. If then a 

 multitude of identical slits are cut into the screen at equal intervals 

 side by side, a new periodicity is superposed upon the periodicity of 

 the waves, and out of the interaction of these two there come diffraction- 

 patterns much more sharp and striking than any which a single 

 aperture, however shaped, is able to produce. These will be considered 

 in the following chapter. 



