Updegraff — Flexure of Telescopes. 247 



glass south, are not equal, the value of a' deduced by this 

 method will be illusory. 



The demonstration of the rule that the astronomical 

 flexure of a telescope in a vertical plane varies as the sine of 

 the zenith distance is as follows:* — 



It was showu above that the force tending directly to bend 

 either half of a telescope tube varies as the sine of the zenith 

 distance. If a weight W at one end produces a linear deflec- 

 tion H when the tube is horizontal, the deflection at any 

 zenith distance z is assumed to be Haiuz. An equal weight 

 at the other end which produces a horizontal deflection of //', 

 gives at the same time a deflection of ^'sin(180° — 2)= H' 

 sing. The ditterence of these deflections is (H — H') s'mz and 

 the astronomical flexure is 



sir -1 /^ - ^' 



5>Ji Sins; , 



\ 21 J 



in which 21 is the focal length of the telescope. 



Astronomical flexure may be produced in two ways by the 

 second cause given above: — 



(1) Bending due to moments of compressive and tensile 



forces. 



(2) Displacement of the neutral surface f by the com- 



pressive and tensile forces. 



We proceed to develop by means of the commonly accepted 

 theory of the elasticity and resistance of materials an expres- 

 sion for the astronomical flexure of telescopes due to the 

 bending moments of the compressive and tensile forces. 



* See Chauvenet's Spher. & Tract. Ast., Vol. II., p. 303. 



t In the theory of the resistance of materials, the terra neutral surface 

 signifies that surface In a bent beam which separates the part of the beam 

 subject to a tensile strain from that subject to a compressive strain. This 

 surface is supposed to pass through the centers of gravity of the cross-sec- 

 tions of the beam; and the term neutral axis, is applied to the line in the 

 neutral surface joining these centers of gravity. (See Weisbach, Coxe's 

 Trans., pp. 410-41.S.) 



