Updegraff — Flexure of Telescopes, 245 



actual position of the line of collimation. Let us suppose that 

 the two halves of the tube (before bending) are symmetrical 

 in form, density and elasticity with reference to the point C\ 

 and let the weight Whe the same at both ends. Then for 

 the upper half of the telescope, the force tending directly to 

 bend the tube is Wsmd and the other component of the 

 weight TF is a compressive force Wco^d acting in the direc- 

 tion OC. The direct bending force for the other half of the 

 tube is W^'\Tid and there is also a tensile force H^cos^ act- 

 ing in the direction CE. The compressive force Wco&d and 

 the tensile force of the same amount tend to shorten and to 

 lengthen the upper and lower halves of the tube respectively 

 by an inappreciably small quantity. There are, however, two 

 causes which may give rise to astronomical flexure under the 

 conditions above specified : — 



(1) Non-homogeneity of the material (usually brass) of 



which the telescope tube is made. 



(2) The existence of a compressive strain on the upper 



half of the tube and of an equal tensile strain on 

 the lower half. 



We take as the normal or undisturbed position of the line 

 of collimation that assumed when the telescope is pointed to 

 the nadir. Little or no flexure is to be expected from either 

 of the above causes when the telescope is directed to the 

 zenith. Flexure due to the first cause would probably be a max- 

 imum when the telescope is horizontal, and there will be two 

 values of the horizontal flexure — one for object-glass north 

 and another for object-glass south. When the telescope is 

 horizontal there can be no flexure due to the second cause 

 since then d = 90° and TFcos^ = 0. Flexure in azimuth 

 can arise only from non-homogeneity of the material of the 

 tube and horizontal axis and must be very small in well made 

 instruments. 



The combined flexures of the telescope and the divided 

 circle are sometimes treated as follows : — 



Let Az = the flexure at zenith distance z; then, we may 

 assume 



