420 Count Paul de Saint-Robert on Atmospheric Refraction, 

 and at the same temperature. Wherefore 



a=p- 

 2ct 



l--£ 



It is to be remarked that the method by which terrestrial 

 refraction is usually calculated implies a decrease of density 

 which agrees tolerably well with that resulting from Mr. Glai- 

 sher's observations. To show this, let us recall the expression 



2k 



© 



d6=— . ■ A1 r dv 

 l + 4kp 



of the element dO of the total refraction*. 



In geodesy it is assumed that the refraction is proportional to 

 the angle at the earth's centre, or that we have 



A6 = 0-6 o = Mv, 



M being a constant. A0 is the angle contained between the 



tangents to the extremities of the arc intercepted between the 



object and the observer's eye, and is equal to the sum of the 



angles which the said tangents make with the chord of the arc. 



As the curvature of the luminous trajectory will vary but little 



in a small extent 3 the two angles are considered as equal ; so that 



Ad 



-5- will be the refraction at each station. The amount of ter- 



restrial refraction is then nearly 



Ae M 



2- = IT' 



It is generally assumed that 



M_J_ 

 2 ""12* 



Let us now inquire what law of decrease of density in the 

 atmosphere this manner of operating presupposes. 

 In order that we may have 



® 



it is necessary that 



-2L— -=M ; 

 1 + 4kp ' 



_ 2kd P _ M ^ 

 l+Up" r' 



By integrating from p and r to p and r, we shall get 

 * Laplace, Mecanique Celeste, vol. iv. p. 2/7. 



