MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



175 



the surface of the atmosphere, and therefore they have no effect to alter the velocity 

 of the light parallel to that surface. Thus 



and 



V sin w ■=■ \J sin w\ 



sin ta 



sin ra^ 



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 light after refraction to the velocity of the incident 

 light ; which ratio, being independent of the direction of the incident light, is con- 

 stant for all light that falls upon the atmosphere with the same velocity. 



What has been said of an atmosphere supposed homogeneous is next to be applied 

 to the real atmosphere of the earth, the density of which decreases continually in 



ascending. The sphere M O N of which C is the centre, representing the earth, let 

 SABO be the trajectory described by light emanating from the star S, in its pas- 

 sage through the atmosphere to the earth's surface at O : through any two pgints of 

 this curve, A and B, draw spherical surfaces concentric to the earth, the distances 

 A C and D C from the common centre being r -\- dr and r. Representing by f the 

 density of the air above the spherical surface at A, let ^ + ^ f stand for the density, 

 supposed uniform, of the stratum between the two surfaces at A and B : and it is to 

 be observed that, though A D = t? r is an infinitesimal, it is nevertheless to be ac- 

 counted infinitely great when compared to the insensible distance at which the mole- 

 cular action of the air at A ceases to act : from which it follows that the refractive 

 power of the stratum upon light which entei*s at A, is exactly equal to the refractive 

 power of a homogeneous atmosphere, supposing the density ^ -\- d^to extend un- 

 varied to the earth's surface. Now if y denote the velocity with which the light 

 moves in the trajectory at x\, the refractive power of the air above the stratum will 

 diminish y- by the quantity 2 (p (f ) ; for it is obvious that the refractive power of the 

 air above the spherical surface at A, is equal and opposite to the refractive power of a 

 homogeneous atmosphere within the same surface and of the density f : on the other 

 hand the refractive power of the stratum will augment u^ by the quantity 2 ip (f -f- d^ : 



