310 The Rev. R. Murphy oji Atmospheric Refraction. 



my doubts public, and shall be thankful if by doing so I should 

 be so fortunate as to elicit an explanation from some of your 

 more experienced correspondents. 



. I am, Sir, your obedient Servant, 

 January 1, 1842. S. EarnSHAW. 



XLVI. On Atmospheric Refraction. J5j/ ^Ae iJew. R. Murphy, 



§ 1. Hypothesis, 

 ^I^HE refracting power of the atmosphere is a function of its 

 '■ distance from the centre of the earth, and tends in the 

 direction of that centre. The curve described by a luminous 

 ray in its passage through the air is governed by the usual 

 laws of the trajectories of bodies acted on by centripetal forces. 



§ 2. — Let the velocity of light entering the atmosphere = I, 

 and V that at which it arrives at the earth's surface. 



Let r be the distance of a point in the trajectory of the ray 

 from the earth's centre, and (f) (?) the force acting then on 

 light ; we have 



t;2 = I +2/,<f>(r); 



the limits of r are 1 (the earth's radius) and \-\-h {h being 

 the height of the atmosphere). Hence v^ is constant for all 

 incidences, and we may put u = 1 + ?« where ?« is a certain 

 constant. 



§ 3. — Let 9 be the angle made by the radius vector drawn 

 to the earth's centre when the ray enters the atmosphere with 

 that ray, and z' be the apparent zenith distance. 



The perpendicular on the ray as it enters the atmosphere 

 drawn from the earth's centre = {\ + h) sin fl, and that as it 

 enters the eye = sin z'. These are inversely proportional to 

 the corresponding velocities, viz. 1 and 1+m; hence 

 (1 + 70 • sin 6 = (1 + m) sin s'. 



^ 4. — Let the observed body be the moon, at a distance a, 

 reckoning in radii of the earth, from the centre. 



Let z be its true zenith distance seen from the earth's centre, 

 corresponding to the apparent zenith distance s'. 



Let p be the moon's parallax at that zenith distance, and 

 let w be the angle at the moon subtended by the bent trajec- 

 tory of the luminous ray. Then p + w is the angle at the 

 moon made by the I'adius vector from the earth's centre with 

 the issuing ray, which being the same that enters the atmo- 

 sphere to reach the eye, we have 



a sin (p + w) = (1 + h) . sinfi. 

 • Communicated by the Author. 



