6 MR. AIRY'S SUPPLEMENT TO A PAPER "ON THE THEORETICAL 



and therefore 



+ Sin R. :^ ; 



and the whole intensity of light is represented by 



/I 1 ^\ /*/sinw\2 , „ /» /cosw.8inw\2 . /^cosw.sin^w 



the limits of integration being ± go. The last term, changing sign when w changes 

 sign, evidently makes its definite integral = : the two former may be put in the 

 shape 



/I 1 T>\ /*/sin«^\2 ,1 ^ r /8in2w\2 



Uy* (^^y from — 00 to + oo be = S, theny* (^^||r) ^^^™ ~" gd to + go is also 

 = S, and the expression becomes 



yY ~ Y cos Rj • S + Y cos R . S or -g- S, 



which is independent of R. The total light, therefore, is independent of R, or is 

 equal at all points ; and therefore no bands are produced. 



2. But if —I— J though small, is not exceedingly small, the principal impression may 



be made upon the eye by the central patch of light from each source, included be- 

 tween the values w =: — 180°, w = -{■ 180°; while those parts of the light which ex- 

 tend beyond the central patch may be in fact aggregated with the central patches of 

 light from the sources at a small distance on each side. And if the amounts in the 

 central patches from different sources are unequal, while the whole amounts from the 

 different sources are equal, it is evident that a bright central patch from one source 

 may be combined with bright detached parts from another source, while a fainter 

 central patch from that second source may be combined with an insignificant detached 

 part from the first source, and thus the whole inequality of light may be double the 

 inequality of the central patches. Now the amount of the light in the central patch, 

 as we have found, is greatest, and represented by 7234, when R = or = 2 w t, and 

 is least, and is represented by 65 17, when R = t or = 2 ?z -f- 1 . t. The difference of 

 these is y^th of the whole ; and therefore the difference of the whole light on each 

 part of the retina, formed by combining the central patch formed by one source with 

 the detached light formed by another source, will be nearly ^th of the whole. This 

 inequality of light is amply sufficient to form conspicuous bands. 



The bars thus formed depend upon nothing but the changes in the value of R ; it 

 is wholly indifferent whether R increases or diminishes towards the side on which b 



