Count Paul de Saint-Robert on Atmospheric Refraction, 417 



Let V be the angle made by the major axis of the hyperbola 

 with the observer's vertical line. The angle V, as well as v, is 

 measured from the vertical line in the direction that causes the 

 radius-vector to increase. Thus 



<£ = 180°-(V-iO; 



therefore 



A(e 2 -1) 

 r = 



1 — ecos (V — v) 



The value of V is found by the condition that when r=r Q , 

 v = 0. Then 



_, A( e *-1) 

 1-ecosV' 

 whence 



tanV= sing ° C0S /° ; 



ccar — sir a 



and putting, for the sake of brevity, 



e aar n — sin 2 6. 



we shall find 



~~ 1 —e cos V ~" sin 2 O cos V 



l-j-CcosV-Ccos(V^) 



for the polar equation of the luminous trajectory, by which we 

 can know the radius-vector and its inclination to the vertical line 

 for any point of the trajectory. 



In order to have the astronomical refraction corresponding to 

 a given zenith-distance, it is necessary first to derive from this 

 latter formula the angle at the centre corresponding to the radius - 

 vector 



r=r + \ 



of the limiting layer of the atmosphere ; afterwards to calculate 

 the angle v' made by this radius with the last element of the 

 luminous trajectory, by the formula 



. ?>> sin n /- — 7 — 



smi/= -^ -Vl + % 



That done, the refraction R will be given, as may be easily seen, 

 by the expression 



R = v + v'-6 . 



Applying these formulae to the calculation of the horizontal 

 Phil. Mag. S. 4. Vol. 27. No. 184. June 1864. 2 E 



