Updegraff — Flexure of Telescopes. 257 



2»r(i)L ^^"^^ 



^ 140 V 2 / 1820 V 2 / • • • • ' 



w,^ r 3/u, \3 

 2'/Xi) L ^^ 



and 



X 



-? r , 9 /t/,\2 81 /u\* 13851 /t/.\6 1 



~ 8 "^ |_ "^ 28 \ 2 / "^ 560\ 2 / "^ 203840 \ 2 / ' ' ' J* 



Substituting this value of X in (33) and reducing we get 



3 / 2 23 43 \ 



J, =gaZnan^ ^1 +^a?3+ 2ao«^ + 1820^'^ ' * )' ^^^^ 



And in the same way we get from Eq. (31), for the lower 

 half of the tube. 



3 / 2 23 43 \ 



j; = ga^nan(? /I — yaZ'+ 2»o"^~" 1820^'^ * * * ' )' (^^) 



In the case of any telescope of ordinary construction a?' is 

 a very small quantity and these series converge very rapidly. 

 Therefore the deflections J^ and A\ may be accurately and con- 

 veniently computed from Eqs. (34) and (35). Subtracting 

 we get 



3 /4 43 \ 



J^ — J,' = ^ alHsLud lijttl^ -\- ^:^a^I^ . . . j, 



3/43 \ 



or J, — J,'=^amand\l + ^a^l^ . . . . j. (36) 



