Updegraff — Flexure of Telescopes. 253 



the flexure curves for the upper and lower halves of the tube 

 respectively, 



El~ = — - wxy cosd — - wxhmd, 

 dx^ 3 2 



(Py 1 1 



EI^2 ~ "I" ^ wxycoad — „ i^x^sin^. 



1 wcos^ 1 tos'md 



If we put a = o jpr and o = ^ pr^ these become 





= — axy — bx^f (19) 



— = 4- axy — bx\ (20) 



It is evident on inspection that Eq. (19) is satisfied by the 

 relation 



y = --x. (21) 



which is a particular solution, so called. 



If to this value of y be added that given by solving (19) 

 with the term bx"^ put equal to zero, the sum put equal to y 

 will, according to a well known theorem of Differential Equa- 

 tions, be the complete or general solution of (19). We 

 have now to solve the differential equation, 



— = — axy. (22) 



dt dy dt 



Putting y = e^ and ^— = w, we have, -p- = e'-^ — 



and 



d?y_ _ fdt^ \2 ^dH 

 d7? ~ \dx / dx^^ 



and Eq. (22) becomes 



du 



dx + ^' = - «^' 



