130 
Proceedings of the Royal Society 
of uniform density, tlie path of the light would be a series of 
straight lines, AB, BC . . . GS ; refraction taking place at each 
junction. 
If 0 be the earth’s centre, and AB the path of the light in the 
first layer, the angle O AB, called by astronomers the nadir-distance. 
is the incidence of the light upon the surface at A, while OBA 
measures the incidence of the same light upon the surface at B ; 
and according to the most elementary laws of geometry, we have the 
proportion 
sin OAB : sin OBA : : OB : OA . 
If now be the indices of refraction for the first and second 
layers, the refraction at B must, according to the well-known law, 
be such that- 
sin OBA : sin OBC : : : /Xq , 
so that, on combining the two ratios, we have 
sin OAB : sin OBC : : OB . : OA . , 
which proportion may also be represented by the equality 
sin OAB . OA . sin OBC . OB . /Xj . 
The analogous equality holds for the path BC, and so along the 
whole course of the light, wherefore the continued product of the 
sine of the incidence, the radius vector and the index of refraction, 
is constant all along ; that is to say, the equality 
sin OAB . OA . /x^ ^ sin OGS . OG . 
