Updegraff — Flexure of Telescopes. 251 



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

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



J-J'= i?*^Man^(l-f ^^4Z* ....), (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 



2 «/TF\2 



^' = 15 K^) «^°2^' (^«) 



from which the diiference of deflection may be computed with 

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

 is always positive for direct observations since 2d 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 — A' 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 heuce will not be eliminated 

 by taking the mean of observations direct and reflected. 

 While it will not in theof^ 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 A — A' given by 

 (16) we have. 



h = sin 





The astronomical flexure varies directly as W^ and in- 

 versely as (Eiy^ 



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



F W 



