Mr. Ivory on the Theoi-yofthe Astronomical Refractions. 287 

 consequently 



rf.S6=-==f:L.- (OS 



but in this formula S fl must be conceived to increase from the 

 surface of the earth to the top of the atmosphere. 



In applying the last formula it is necessary to have a value 

 of y. Draw O L to touch the curve at O, and C N perpen- 

 dicular to O L : put up' ' for the density of the air, and the 

 velocity of the light at O; also?/' for the perpendicular C N, 

 a for C O the radius of the earth, and fl for angle CON, 

 which is the apparent zenith-distance of the star : we shall 

 have Area ABC dz 



dr = 7/T^-^="^-^ = 



and because the curve is described by a centripetal force 

 tending to C, the value of u x?/ will be the same at all the 

 points of the curve; wherefore 



u X y = v' X 3/' : 

 and 



y =y X — = a sm 9 X — . 



Now, according to what was before shown, 



wherefore 



VI +2<j^(p'); 



^J\ 



+ 2<f(p) ^^-^ 



By substituting this expression in the differential of the re- 

 fraction, the problem will be reduced to an integration. 



The equations that have been investigated are perfectly 

 general, and will apply in any constitution of the atmosphere 

 that may be adopted. It has been thought better to consider 

 the manner in which the forces act, than to employ functions 

 with peculiar properties to express the molecular action. 

 When the light in passing through the atmosphere arrives 

 at a surface of increased density, it receives an impulse which 

 may be considered as instantaneous ; and this impulse beinw 

 distributed over the breadth of a stratum of uniform density, 

 ascertains the centripetal force tending to the earth's centre, 

 by the action of which the trajectory is described. 



[To be continued] 



