ihe fixed Supports of flexible Stilslance$. 141 



fiHcrums, y and z, the strain at the distance x from the niidd'e 

 will be, for the portion next the middle, y[2a — x) -\-z{Aa — x) — 



— (5a— a;)'; the indination 1ayx — —yx''-- -{- Aazx— -^■x.x'^ —■ 



25 5 1 . 1 



-^ a'^x -\- — nx' ra:'; and the ordinate ayx" r yx' -f- 



2azx^ tZX' — a-a;^+ — ax' — T^rx', which must vauisli 



() 4 O it 



when X — 2a, and = 4a^y a^y + Sa^z— -- a^z — 25a* + 



— a^—-a*=-y+ -^-z —a, and 8y + 20a; = 57a. In 



the next portion the pressure y is not concerned, and the ex- 

 pression for the nclination becomes 4a;2X— --;i;a;'— — a -a; + 



—ax^— ■x^ + l; and for the ordinate 2azx' — — zx^— — a^x* 



a D 6 4 



5 ] 



+ —ax'— -rx^ + ^aj + c: here i must be determined from the first 



t» 'i-i- 



portion, the final indination of the one coinciding with the initial 

 inclijiatioii of the other ; and making in both expressions x=2a, 



8a'2; — 2a'2; — 25a' + 10a'— ~a^ + b= 4a'-y — 2a'y + 8a^z — 



2fl'2;— 25a' -r ICa'— —a^; consequently l = 2a^y: then the 

 ordinate vanisliing at the beginning of the second portion, when 



4 "0 " 



x = 2a, we have 0=:8a'2;— — o'« — 25a-* + —c-* — ^a-* + 

 4a^y + c; and when x~4a, and the ordinate once more vanishes, 

 = 32a'z— -|^a'z-100a-t + -l|-'a+- ^a^ + Sa'?/+ c; and 



by subtraction, 24a'a;— ~a^z — 75a'* + ~t^^ — T"'^* "^ ^^^y 

 ■=0-=: -^z :r-a -\-4y; whence subtracting 4y-{-l0z a, 



1 V 59 59 52 



we have 0= — z ra, and z= ^<^i consequently y = — ff, 



to 



and for the pressure supported at the middle, there remains —a. 



If we now suppose the fulcrum at the middle to be removed, 

 the equations for the second j)art of the figure and for the in- 

 clination of the first part remain unaltered, and we have also 



4 (• 



y + 2 rs 5c, and 4y + 4z — 20o, which, subtracted from —x — 

 -T- a + 4y = 0, leaves -^z — -«=0, and a5=r— -a, and y = 



105 



D. Ill 



