54 SMITHSONIAN MISCELLANEOUS COLLECTION'S, VOL. 5 I 



IV. SECOND APPROXIMATE VALUE OF II J ACCURACY OF THE RESULT 



By reason of equation (15) the triangles bed and amd, fig. 5, 



are similar to each other, wherefore for the point of symmetry a 



we have 



, am , TT 2 h ^ 



be = cd or H 



dm R + h 



and substituting the values from equations (4) and (19) we get 



I" n' - 1 



L 57.3" 



H = 2 R \ — (l - A) - n' 



= about 189.0 km. . . . (20) 



as compared with 192.6 in equation (7). 



The agreement of these two values shows that the error made 

 by equating the distances in equation (14) is without important 

 influence on the result of the computations; that therefore in 

 fact the distances of the points of symmetry from the boundaries 



of the atmosphere are nearly equal to each other when - = 57.3" 



and «' — 1 = 0.000282. 



But of the two formulae (7) and (20) for H the first is more exact 

 because the curvature of the beam of light through the zenith de- 

 parts but infinitesimally from a circular arc. 



As to the numerical determination, H = 192.6, the assumption 

 that the ratio of the refraction to the zenith distance (57.3") is cor- 

 rect to within 0.001 part of itself makes A < 0.001 and this would 

 lead to an error of a few kilometers in the determination of the 

 height of the atmosphere. 10 



V. CONCLUSIONS 



On the basis of the preceding determinations it seems natural to 

 attempt a new development of the differential equation (9) for 

 astronomical refraction. 



The law of diminution of refractive power with altitude may with 

 great probability be 



»' - 1 



P 1 



10 1 have recently found that the numerical determinations of - , W = 1 , 



c 



and H really do need important corrections. 



