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



In like manner (4) becomes, 



»2 , 1 (AX _ e - qx X 



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 A for the deflection of the upper end 

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

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



pi (i e i<H_ e -iQl \ 



J =?(g • .«+,-« -')• (10) 



rf / I e ql e -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, 



h = sin 



M- w 



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

 lows. Developing the exponential functions e iql , e~ iql , t ql 

 and e~ ql in series, substituting these values and also the values 



of p 2 and q 2 in -t_, (10) and (11) become after reduction, 

 q 2 



J = h H3t ™ d { l + h 2l2 + m« Hi +Wo^ ■ • • ■)■ 



(13) 



A >=lq 2 inznO(l-lq 2 P + gqV-^qn« . ... .). 



(14) 



Since q' 2 = — J^— , q 2 l 2 will always be a small quantity in 



case of a telescope of ordinary construction. Therefore 

 these series converge rapidly and the absolute deflections A 



