258 Trans. Acad. Sci. of St. Louis. 



Only the first term of this series will ever be appreciable. 

 Neglecting the others and putting for a its value and for wlj 

 W^, we have 



which has the same form as Eq. (16). 



jj — a\ is always positive for direct observations, is a 

 maximum for 6 = 45°, and is zero for ^ = 0° or 90°. 



If in Eqs. (34) and (35) we put d = 90% then 



^^ = ^'^ = 8 wr 



which is the well-known formula for the deflection of a hori- 

 zontal, uniformly loaded beam, fixed at one end and free at 

 the other.* The formula for the astronomical flexure is 



h^ = sin 



-1 



Z* /WA' 



168 



[ml " 



sin2(9 



(38) 



If in the difl'ereutial equations (19) and (20) the value of 



, ,, , Iwcosd ■ . 1 nl wcoad , , , , 



a had been made — — — - instead of- -— — ^ we should nave as 

 2 JSI 3 UI 



3 



the coefficient of the right hand member of (37) in- 



= ^ M12 



stead of While (37) gives theoretically the law of varia- 



tion with the zenith distance of the astronomical flexure 

 due to the weight of the tube it can hardly be expected to 

 give even a rough approximation to its absolute value. t 



* See Wood's Besistance of Materials, p. 110. 



t For an attempt to deduce from the theory of the elasticity and resist- 

 ance of materials formulae for computing the astronomical flexure of teles, 

 copes, see an article by V. Baggi in the Astronomische Nachrichten Nr. 3285. 



