On Terrestrial Refraction 



547 



from the air. They should be of interest to the surveyor. 

 Further, it seems to the writer th;it the method followed is 

 the only sound one to be used in obtaining astronomical 

 refractions, when our knowlege of the upper atmosphere 

 more complete. 



is 



1. Criticism of the usual assumption that Light Rays in the 

 Atmosphere are Arcs of a Circle. 



Suppose the earth to be a sphere of radius R, surrounded 

 by an atmosphere which is arranged in a spherically sym- 

 metrical way, so that its density is a continuous function of r 

 alone, r being the distance from the centre of the earth. Let 

 (f> be the zenith distance of the ray, i. e. the angle the ray 

 makes with the radius from the centre, after crossing it. 

 Then the following well-known formula connects the re- 

 fractive index, n, of the air with r and <fi : 



nr sin <j> = k (1) 



The value of k varies from one ray to another, but is constant 

 for a particular ray. If the length dl is the differential of 

 the particular ray of light considered, we have at the point F 



(fig. i) 



1 dn . 1 dr . , , d<j> __ n 



Fitr. I. 



(2) 



Let p be the radius of curvature of the light, ray at P : and 

 let 7 be the angle that the radius from the earth's centre 



