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 wl, 

 TFj, we have 



'^-^'^ = k{m)''''''^ ^''^ 



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 



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 ^ 



168 {nil 



(38) 



If in the differential equations (19) and (20) the value of 



a had been made- J,^ instead of _ ^- — we should have as 

 2 JEl 3 UI 



3 

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



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. f 



* 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. 



