Mr. Ivory on the Theory of the Asti'onomical Refractions. 285 



the other surface m n' ; but when it has passed this limit, it 

 will no longer be acted upon effectively by the surrounding 

 air, which will attract it e()ually 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 consi- 

 der the limiting surfaces m n and m' n' as parallel planes, the 

 forces being perpendicular to m ??, and of the same intensity 

 at all equal distances from it. The law of the forces in action 

 between m 11 and 7?z'n' is indetermined ; it may be uniform, or 

 varied in any manner. These things being premised, it ibl- 

 lows from a fundamental proposition of the j)hilosophy 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 n' is to add to the square of the velocity of the light 

 incident at vi 21, an increment which is always the same, what- 

 ever 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 v' for the velocity with which it leaves ?«' n', what is said 

 will be expressed by this equation, 



<p (g) denoting the sum of all the forces between m n and m' «', 

 each multiplied by the space through which it acts, a sura 

 which, in different atmospheres, will vary only when p varies. 



It will be convenient to have a name for the function <p (p), 

 and the most appropriate term seems to be, the refractive 

 power of the air. In using this term, or in expressing by 

 (^ (f) the action of air upon light, it is always supposed that 

 the light passes out of a vacuum into air of the density g. 



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 ct de- 

 note the angle made by the direction of the velocity y with 

 the radius, and ■uj' the angle made by the direction of the 

 velocity o' with it ; then u sin ct and u' sin ot' 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 u' are perpendicular to the surface of the at- 

 mospiiere, and therefore they have no effect to alter the ve- 

 locity of the light parallel to that surface. Thus 



V sin us = v' sin zr', 

 und 



sin •BT v' 



sin •ct' ~ y ' 

 that is, in words, the ratio of the sine of incidence to the 

 sine of refraction is equal to the ratio of the velocity of the 



