174 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



the trajectory in which the light moves, being described by a centripetal force, the 

 determination of its figure will fall under the propositions contained in the second 

 section of the same book. 



Conceive that light falls upon an atmosphere A G K, constituted as Cassini sup- 

 posed, spherical in its form, concentric to the earth, of 

 the same density f throughout ; and suppose that the at- 

 tractive force of the molecules of air situated in the sur- 

 face A G K extends to m w on one side, and to m' ri on 

 the other. Every molecule of light when it arrives at 

 Y. m n will be attracted by the air in a direction perpen- 

 dicular to the surface A G K, and tending to C the centre 

 of the earth ; it will continue to suffer a varied attraction 

 till it penetrates to the other surface m' ri ; but when it 

 has passed this limit, it will no longer be acted upon effectively by the surrounding 

 air, which will attract it equally in all opposite directions. As the attraction of air 

 extends only to insensible distances, in estimating its action upon a molecule of light 

 we may consider the limiting surfaces m n and m' w' as parallel planes, the forces 

 being perpendicular to m n, and of the same intensity at all equal distances from it. 

 The law of the forces in action between m n and m! ri is indetermined ; it may be 

 uniform, or varied in any manner. These things being premised, it follows from a 

 fundamental proposition of the philosophy of Newton, the demonstration of which 

 it would be useless to repeat here, that the total action of all the forces between m n 

 and m' w' is to add to the square of the velocity of the light incident at m n, an incre- 

 ment which is always the same, whatever be the direction in which the light arrives 

 at m n. If we now put v for the velocity with which the light enters m n, and y' for 

 the velocity with which it leaves m! w', what is said will be expressed by this equa- 

 tion, 



t>'2 - u2 = 2 . 9 (f), 



(p (§) denoting the sum of all the forces between m n and m' n', each multiplied by the 

 space through which it acts, a sum which, in diflferent atmospheres, will vary only when 

 g> varies. 



It will be convenient to have a name for the function (p {§), and the most appro- 

 priate term seems to be, the refractive power of the air. In using this term, or in 

 expressing by <p (g) the action of air upon light, it is always supposed that the light 

 passes out of a vacuum into air of the density §. 



A property resulting from what is said may be mentioned. Having drawn a radius 

 from the centre of the earth to the point at which the light falls upon the atmosphere, 

 let B7 denote the angle made by the direction of the velocity v with the radius, and ^' 

 the angle made by the direction of the velocity v' with it ; then v sin nx and v' sin zj' 

 will be the partial velocities of the light parallel to the surface of the atmosphere. 

 Now these are equal ; for all the forces which change v into v' are perpendicular to 



