Prof. Everett on (he Optics of Mirage, 259 



dT dT dT 

 Substituting for -7-, -7-, — their values k cos a, h cos ft, A- cos y, 



we have 



Therefore 





V^T . VrfT VdT 



cosa= j— ; cos/3 = =—: cos 7= r - • 



fz ace fi up A 6 "~ 



To apply these last formulae to the investigation of the curva- 

 ture of the ray at any point, make the axis of x coincide with 

 the tangent (drawn forwards), and the axis of y with the prin- 

 cipal radius of curvature (drawn from the point towards the 

 centre of curvature). Then, in the neighbourhood of the origin, 

 we have, to the first order of small quantities, 



V dT P • 



r- == cos p ~ sin a, = a : 



p ay 

 and the curvature is 



l__du_d_^YdT\_ _Y_dpdT V d*T _ 

 p dx dx Kfjb dy J p? dx dy p, dx dy 



But 



Therefore 



^T _ ,dT 1 n 

 -7- = 0, and -7- = - = T - 



dy dx v V 



1 V d*T Yd /u,\ I dp 



lJL(f±\-L 



p dy\SJ~' fi 



p fju dx dy jjb dy\Vj fi dy' 



and. from our choice of axes, ~ denotes the rate at which a in- 



dy 



creases in travelling towards the centre of curvature. 



Two other proofs will be found in Parkinson's ( Optics/ arts. 



122, 123; the result there obtained, namely 



fjb __d\z dy dfi dx 

 p dx ds dy ds 



being clearly equivalent to that which we have employed, since 



du dx 



-j- and — -7- are the direction-cosines of p. 



as as 



I may also remark that the formula 



d/z ___ d / dx\ 

 dx'~ds~\ fJj ds~)' 



which is usually obtained by a difficult application of the Cal- 

 culus of Variations, can be immediately derived from the princi- 



S2 



