556 



Mr. A. R. McLeudo» 



Formula (29) may be used instead of (26) for zenith 

 distances not exceeding 85°, the error caused by the approxi- 

 mation not exceeding 1". Since « = when <£ = 7r/2, it is- 

 obvious that the assumptions under which (29) was derived 

 are no longer valid when the rays pass close to the horizon.. 

 Formulae (27) and (29) are, of course, subject to the same 

 restrictions as (26), and cannot in an}' case be relied on 

 to within 1" for heights greater than 1 km. 



The reason for the variation of the constant m in (27) is 

 at once apparent. If the radius of curvature of a light ray 



is assumed to be constant, it is, by (15), : — r . Hence 



J yn sm<f> 



m =gn sin <f> ; or, approximately, m =g sin </> . Substituting 

 in (14) the observed values of a at various heights, and the 

 value yu, = '2271, we find the values of q which are contained 

 in Table III. 



Table III. 



h> g- 



km — 



1 „ -1796 



2 „ -1661 



3 „ -1655 



4 ., -1554 



5 „ -1479 



6 „ 1420 



7 ., -1362 



hi. g. 



8 km '1311 



9 -1260 



10 4218 



11 ., -1179 



12 „ -1134 



13 4101 



14 „ -1063 



15 , -1020 



The values of m will thus vary within the approximate 

 limits '18 and *10. In any particular survey there will be a 

 mean value of m, more or less applicable to the totality 

 of measurements made, and the values should lie, us they do,, 

 within these limits. 



5. Curvature of the Ligld Rays. 



Consider that portion, PQ (fig. 3), of a ray which lies 

 between the heights /^_j and A»=Ai_i+ 1, where we suppose 

 hi=i km. (2 = 1, 2, .... 8). Let the length of this part 

 of the ray be U ; let it subtend an angle ji at the centre of 

 the earth; and let </>;_! and fa be its zenith distances at 



