416 Count Paul de Saint-Robert on Atmospheric Refraction. 



curve with the same vertical line ; v' the angle made by the tan- 

 gent with the radius-vector r ; p the density of the air at the 

 assumed point ; 4k the refractive power peculiar to atmospheric 

 air of the density 1 ; and r , O , p the values of r, 6, p at the 

 point where the trajectory enters the observer's eye. 



The determination of the luminous trajectory depends upon 

 the following equation, 



r 8infl _ \Zl+4kp m 



rsinv ' " Vl+Mp ' 



where r sin 6 , r sin v' are the perpendiculars let fall from the 

 earth's centre upon the tangents, making respectively the angles 

 6 , v' with the radius-vector. 



Introducing in it the assumption 



Po \ r I 



and making, for the sake of brevity, 



we get 





1 2a,a 1 l — 2xar 



(rsinz/) 2 sin 2 # r (r sinfl ) 2 

 Now by conic sections we have in the hyperbola 



J L _2A 1 1 



N 2 ~B 2 V" f B 2 ' 



N being the perpendicular from the focus upon the tangent, A 

 and B the semiaxes of the hyperbola. 



Hence the luminous trajectory is an hyperbola whose focus is 

 in the centre of the earth, and whose semiaxes are 



A _ " ar <? B __ r sinfl 



l-2ctar ' \/\-2aar Q 



The polar equation, about the earth's centre, to the luminous 

 trajectory will then be 



A(e 2 -1) 



r= — -.9 



\-\-e cos cf> 



cf) being measured from the vertex, and 



A 2 + B 2 , sin 2 ( 



^=1+^(1-2^), 





