248 



This function we call K{.c,^,?-); it represents tlie deflexion of 

 the beam, loaded by a load 1, which is concentrated at the point §. 

 Putting A := — Q* the function 



=F —— \ sinh Q («— §) — sin q («— i) } 



(the upper sign for .c £. §, the lower for x>_ g) will satisfy all 

 conditions excepted those at llie ends. 

 Assuming 



^(«, §, A) = q= -— - I sinh Q (.1-— i) — sin p (^— §) j + 



-\- A cosh Q {x— i) -{- B sinh 9 (.r — è 'r^^ '^o' P (•'^— 2 + -^ *'" Qi" ^i 0> 



we may determine A, B, C and D in such a way that K{x, 5, A) 

 satisfies the conditions at the ends. This gives 



— A cosh ^qI ^ B sinh ^()l-\-C cos hgl—D siri ^ qI — --y j sinh Qi, -\- sin q^\, 



— Asinh^Ql-{-Beosh^Ql—Csin{Ql—I)cos^Ql= -~ \coshQ^-\-tosQS,\, 



— A cosh \ qI — B sinh { qI-^C cos ^qI-\- D sin ^qI = 



= -— \ sinh Q (l—^) + sin q (l—^) \, 



— A sinh ^ qI — Bcosh^(}l — Csin ^qI-\-JDcos ^ qI=: 



= -^ \ cosh Q (I -§) 4- cos Q {l—é,) \. 



1 



V 



Adding the first and the third of these equations and also the 

 second and the fourth we get two equations containing only A and C. 

 Subtracting the third from the first and the fourth from the second 

 we get two equations containing only B and D. In this way we 

 obtain 



— A cosh h qI -t Ccos^ p' = !«»«^ 2 QlcoshQ{§—^Ql)-{- sin J qIcosq(£—U)\, 



4q' 



— A sinh ^qI — Csin kQl^ — - \ cosh ^ qI cosh q (§— ^ I) -\- cos ^ qI cos q (b— èO !' 



I . , . ♦ 



Bsinh ^qI — D sin ^ ^i = \cosh ^qI sinh q (§— ^ I) -\- cos h qI sin q (f — i /)}, 



4q' 



Bcosh ^qI — Dsi7i ^(fl^^ [sinh | Qlsi7ih q (s~2 — **" a Qlsin p(§— iOji 



4^* 



