208 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER 



index at any point is a function of the distance from a 



fixed point, 



dO C 







dr 



r \Sfji L r z ■ 



-C 2 ' 



From this 



equation 



we o 



ds 

 dr 



btain 



rfju 







^?> 2 fJL Z - 



-C 2 



Let R be the earth's radius, and z the apparent zenith 

 distance at the place of observation ; then if the refringent 

 power of a gas be constant, so that /x 2 — i=/e^, we obtain 



ds _ r V ( i + fed) 



dr */r z (i+icd)— R 2 sin 2 * 



This leads to the following differential equation for deter- 

 mining the intensity of the transmitted light at any 

 point : — 



f~ R 9R 2 



d log I ^ -fi 9 — r\/ i+kT>6 9 » e ur 



v 



dr r / R gB? 



-9- - 



r 2 (i+/dDe ' a e ar ) — R 2 sin 2 * 



III. We might also have the absorbing matter distributed 

 through the medium in such a manner that the same value 

 for the density would recur. Suppose, for instance, that 

 the colouring-matter is distributed in parallel layers, and 

 that the law of density is given by the equation 



d=m — n sin t, 



where t denotes the distance from some plane, and m 

 and n are constants, m being greater than n } the initial 

 density will be m, and the value will recur whenever sin t 

 vanishes. The density will have a maximum value m + n 



whenever t is of the form 2v7r + 3— , where v has any 



