6 
MR. AIRY’S SUPPLEMENT TO A PAPER “ON THE THEORETICAL 
and therefore 
/ sin v)\ 2 n { R\ /l 1 „\ /sinw\ 2 , „ /cos w . sin w\ ‘‘ 
) «= 0 S 2 (w> - 2 ) = [r - T cos R) ) + cos R y ) 
+ sin R . 
cos uo . sirr w 
w* 
and the whole intensity of light is represented by 
/I 1 r /sinw\2 /* /cosw.sinw;\2 . p 
It-2-cosRJ / Air ) + COS R / { ^ ) + sm R / 
cos w; . sirr w 
w 2 ’ 
the limits of integration being + oo. The last term, changing sign when w changes 
sign, evidently makes its definite integral = 0 : the two former may be put in the 
shape 
/I 1 Tn\/V s i nw \ 2 . 1 r /sin2w\2 
(t - ¥ cos R l/„ 1-5-) + ¥ cos R 7 2 „ \ / • 
If/* from — qo to + cc be = S, then^ fr° m — go to -f c© is also 
= S, and the expression becomes 
cos r) ■ S + -^-cosR.S or-^-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\e 
2. But if — , 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 = 0 or = 2 wtt, and 
is least, and is represented by 6517, when R = t or = 2 n + 1 . r. The difference of 
these is -j^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 
