IN PASSING THROUGH THE ATMOSPHERE. 
237 
material particles which it contains, without reference to their distribution). Conse- 
quently 
— Q . s . %dr. sec v' (1.) 
Q being a constant quantity. 
36. ii. To find the differential of refraction . — In fig. 4. 0 is the angle formed with 
the zenith by any element A' A of the path of a ray A 1 AO refracted in the atmo- 
sphere. Hence the variation of $ or dO is the differential of refraction. Now 
0 = v -f- v 1 (2.) 
dO — dv + dv' (3.) 
37- iii. When light is refracted at successive concentric surfaces of varying density, 
the flexure of the ray at each is the differential element of refraction. Thus in fig. 5. 
it is evident that the course of a ray being represented by the polygon ah c d e, the 
Fig. 5. 
a 
flexure at the common surface of each successive stratum 4', &c., due to the 
inequality of the angles of incidence and refraction, measures the actual deviation of 
the ray from a rectilinear course*. Hence 
refraction = 4* + 4 1 ' + 4 1 " + &c., 
and hO = -^ (4.) 
38. iv. When light passes from one part of a medium whose density is §, to another 
part of the same medium whose density is f 
sine incidence : sine refraction = \/ 1 + h §' : y / 1 + k § , 
k being the refractive power of the medium. This optical principle is derived from 
experience. 
Let §' = g + S g>, then 
sin incidence \/\ + + £8^ 
sin refraction f\ = +Tcl> " 
* I specify this, because at first sight it would seem as if the flexure in question taking place round a normal, 
which is itself continually changing, would not express the due deviation. 
