Intelligence and Miscellaneous Articles. 407 



by a quantity which, in the mean, is a fifteenth of the terrestrial 

 arc extending from the point where the ray enters to the point at 

 which it arrives. Thus for a horizontal path of 1852 metres, which 

 is equal to a minute of an arc on the terrestrial globe, the deflection 

 or refraction of the ray would be -^ of a minute, or 4". 



There are three things to be considered in this question ; — 



1. The trajectory of the ray is a circle. 



2. There is a constant ratio n between the quantity by which the 

 ray is inflected, and the terrestrial arc comprised between the point 

 of entrance of the ray taken to be horizontal, and its point of 

 arrival. Then let s be the angle at the earth's centre comprised 

 between these two points, and r the refraction, we have 



r „ B 



7 O m -76 (l+af)UO m -76d M / 



Here R is the mean radius of the earth ; B is its atmospheric pres- 

 sure reduced to zero ; a is the coefficient of expansion ^H of the 

 air for 1° C ; d is the density of mercury as compared with air taken 

 at zero ; and (what is new and important) M is the height in metres 

 corresponding to a diminution of one degree on the Centigrade scale. 

 This formula, expressed numerically, becomes 



W= -A___L_ io-2345-^U. 

 m -76(l+a0 2 l M J 



Several remarkable conclusions may be deduced from this, relating to 

 the physical constitution of the atmosphere. 



3. If the ray does not travel horizontally, but is inclined to the hori- 

 zon at an angle i, the atmospheric refraction diminishes in the ratio 

 of cos i to unity ; but then the path of the ray being greater than its 

 horizontal projection in the ratio of unity to cos i, a compensation is 

 established ; and calling s the angle at the centre of the earth com- 

 prised between the signal and the observer, we have, as before, 



r -=„=^ L_{o-234S-«^l. 



s O m -76 (1 + aO* I M J 



There would be no refraction, and the ray would travel in a right 



Cm. Q£7 



line, if 0-2345 m~=°> which S ives M=29 m -3. Thus if the 



temperature of the air sank 1° for 29 m 3, there would be no refrac- 

 tion. On the other hand, taking B = O m, 76 and £=0, we have 



6 m -867 

 ,=0-2345- - W ; 



taking this quantity as equal in the mean to -J-g-, or - 0G67, we have 

 about 41 metres for M : all these quantities are much less than 200 

 metres, which is the height necessary to be traversed to have a dimi- 

 nution of one degree of temperature in the higher layers of the 

 atmosphere. 



The coefficient n varies from 0500 to 0000 : it can even become 

 negative, which corresponds to the case of mirage whenever M is 

 less than 29 m- 3. We shall see afterwards the great influence which 



