250 Trans. Acad. Sci. of St. Louis. 



In like manner (4) becomes, 



Equations (8) and (9) are the equations of the neutral 

 axes of the upper and lower halves of the telescope tube re- 

 spectively. If we put J for the deflection of the upper end 

 of the tube and J' for that of the lower end we have, since in 

 (8) and (9) y = A and A' for x = I, 



7)2 / 1 f.ql p-ql \ 



The deflections of the upper and lower ends of the tube 

 may be computed from (10) and (11), and then the astro- 

 nomical flexure will be given by the formula, 



A=sm-(^^). (12) 



But it is more convenient to transform (10) and ( 11) as fol- 

 lows. Developing the exponential functions e'^\ e~*^\ e^' 

 and e~'' in series, substitutino; these values and also the values 



of p^ and q^ in ■^, (10) and (11) become after reduction, 



1 / 2 17 fi2 \ 



(13) 



1 / 9 17 fi2 \ 



,^-qWi..e{l--^qV+^^q'P-^^,fl^ . . . .). 



(14) 



TFcos^ 

 Since q^ = — :^^ — , q^P will always be a small quantity in 



case of a telescope of ordinary construction. Therefore 

 these series converge rapidly and the absolute deflections A 



