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522              ON  THE  SCATTERING  OF  LIGHT  BY  SPHERICAL  SHELLS              [427
when the integration is taken for a complete sphere of radius R. If m is moderately great, that is, if R be a large multiple of \ the first term on the right of (11) preponderates, and we may use approximately
_ ^x    ......................... ^    '
Thus, if (K- 1) has no constant part,
,                  =    flfrX   ^
'    2&cos^%    4 cos | ^ ...................... ^    '
The relation between the wave-length of the structure (A) and that of the light is expressed by
A = |X/cos 1^ ................... . ........... (14)
" It seems probable that a structure of this sort is the cause of the remarkable colours, variable with the angle of observation, which are so frequent in beetles, butterflies, and feathers.Now