140 Lord Bayleigh on the 



that the whole difficulty of the problem depends upon the 

 curvature of the earth. If this be neglected, that is if the 

 strata are supposed to be plane, the desired result:' ollows at 

 once from the law of refraction, without the necessity of 

 knowing anything more than the conditions of affairs at the 

 surface. For in virtue of the law of refraction, 



fju sin = constant ; 



so that if be the apparent zenith distance of a star seen at 

 the earth's surface, and 86 the refraction, we have at once 



/*o sin 0= sin (0 + 80), (19) 



from which the refraction can be rigorously calculated. If 

 an expansion be desired, 



80= sin 86= tan (p — cos 86) 



= (^ o -l)tan0{l + i(^o-l)tan 2 0} . . (20) 



is the second approximation. 



When the curvature of the earth is retained, so that the 

 atmospheric strata are supposed to be spheres described round 

 the centre of the earth, the appropriate form of the law of 

 refraction is 



fip = constant. 



Thus, if A be the point of observation at the earth's surface 



where the apparent zenith distance is 0, and if the original 

 direction of the ray outside the atmosphere meet the vertical 

 OA at the point Q, 



/kq. OA, sin 0=OQ. sin (0 + 80) ; 



or if OA=a, AQ = c, 



^ o asin0=(a + c)sin (0 + 80). . . . (21) 



