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 t 

 W v we have 



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



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

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



If in Eqs. (34) and (35) we put 6 = 90°, then 



J * = J 'i = 8 Bt 



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 ^r-l sin2# 



(38) 



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



.j. i 1 ?^COS^ . . , - 1 WCQS0 , , , , 



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



2 El 3 EI 



3 



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



v J 112 



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

 84 v ' fi J 



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. 



