Updegraff — Flexure of Telescopes. 251 



and J' may be conveniently and accurately computed by (13) 

 and (14). Subtracting (14) from (13) we get, 



J-A'= i qH^ tan^(l + ^ qH' ....), (15) 



in which only the first term of the series will be appreciable. 

 Neglecting all terms except the first and substituting for q'^ 

 its value, we have 



J-J'=^?^(^) sin2^, (16) 



from which the difference of deflection may be computed with 

 convenience and accuracy. From (16) we see that J — A' 

 is always positive for direct observations since 26 can never 

 be greater than 180°, and hence the flexure of the upper half 

 of the tube is always theoretically greater than that of the 

 lower half. 



For observations by reflection J — J' is negative. The 

 position of the telescope when pointed to the nadir being 

 taken as the undisturbed position, the flexure is zero at the 

 zenith and the horizon and is a maximum at zenith distances 

 of 45°. 



Flexure of this kind will diminish both zenith-distances and 

 nadir-distances (as measured) and hence will not be eliminated 

 by taking the mean of observations direct and reflected. 

 While it will not in theory give rise to a discordance Reflected 

 minus Direct it does not follow that it will not so do in 

 practice. 



Substituting in Eq. (12) the value of J — A' given by 

 (16) we have. 



h = sin 



I* / TF\2 



sin 2(9 



15 \Wl) 



(17) 



The astronomical flexure varies directly as W^ and in- 

 versely as ( Biy 



If in Eqs. (13) and (14) we put 6 = 90° we get, 



I' W 

 A = A' = -^-^, (18) 



