464 



PHYSICAL SCIENCES. 



to twice the angle ZAP. Again, S and S' being the sun at his greatest 

 distances from the equinoctial AQ when in the solstices, the sum of 

 his true zenith distances, Z A S, Z A S', is equal to twice the angle 

 Z A Q. Consequently, the four true zenith distances, when added 

 together, are equal to twice the right angle Q A P ; that is, they are 

 equal to 180. But the observed or apparent zenith distances are less 

 than the true on account of refraction; therefore the sum of the four 

 apparent zenith distances is less than 180 by the whole amount of the 

 four refractions. 



Fig. 51. 



NOTE 191, p. 155. Terrestrial refraction. Let C, fig. 51, be the 

 centre of the earth, A an observer at its surface, A H his horizon, and B 



some distant point, as the top of a 

 hill. Let the arc B A be the path 

 of a ray coming from B to A; 

 E B, E A, tangents to its extremi- 

 ties ; and A G, B F, perpendicular 

 to C B. However high the hill B 

 may be, it is nothing when com- 

 pared with C A, the radius of the 

 " -"^ earth ; consequently, A B differs so 

 little from A D that the angles A E B 

 and A C B are supplementary to one 

 another ; that is, the two taken to- 

 gether are equal to 180. A C B 

 is called the horizontal angle. Now 

 B A H is the real height of B, and 

 EAH its apparent height; hence 

 refraction raises the object B, by 

 the angle E A B, above its real 

 place. Again, the real depression 

 of A, when viewed from B, is 

 FBA, whereas its apparent de- 

 pression is F B E, so E B A is due 

 to refraction. The angle F B A is equal to the sum of the angles BAH 

 and A C B ; that is, the true elevation is equal to the true depression and 

 the horizontal angle. But the true elevation is equal to the apparent 

 elevation diminished by the refraction ; and the true depression is equal 

 to the apparent depression increased by refraction. Hence twice the refrac- 

 tion is equal to the horizontal angle augmented by the difference between 

 the apparent elevation and the apparent depression. 



NOTE 192, p. 155. Fig. 52 represents the phenomenon in question. 

 S P is the real ship, with its inverted and direct images seen in the air. 

 Were there no refraction, the rays would come from the ship S P to the 

 eye E in the direction of the straight lines ; but, on account of the variable 

 density of the inferior strata of the atmosphere, the rays are bent in the curved 

 lines P c E, Pc?E, SwE, SwE. Since an object is seen in the direction 

 of the tangent to that point of the ray which meets the eye, the point P of 

 the real ship is seen at p and p', and the point S seems to be in s and s' ; 

 and, as all the other points are transferred in the same manner, direct and 

 inverted images of the ship are formed in the air above it. 



