Vpdegraff — Flexure of Telescopes. 251 



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

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



A - J '= T5 ^ tan/? ( x + m ^ 4 ••■•)' (15) 



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

 Neglecting all terms except the first and substituting for q 2 

 its value, we have 



2 / W \ 2 



J - J, = i5' 5 bi) sin2 ^' ( 16 > 



from which the difference of deflection may be computed with 

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

 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 A — 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 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 A — A' given by 

 (16) we have, 



h = sin 



JlL ( w \ 



15 \E1) 



shi20 



(17) 



The astronomical flexure varies directly as W 2 and in- 

 versely as {EI) 2 



If inEqs. (13) and (14) we put = 90° we get, 



I 3 W 

 A = A' = ~2~j^j, (18) 



