252 



4. We now suppose y to be the required solution of (1), viz. that 

 solntioii for which i/'^y"' = in the points .r =: and x = l and 

 wliich is continuons in (0,/) as well as its first three derivates; 

 as to y"" it may have a saltus in a finite number of points a,-, 

 which will be the case if q' (.r) has in the points a, discontinuities 

 for which q' (r/, -|- 0) and q {ai — 0) exist. The points a, and the 

 value 5 divide the interval (0,/) into a number of subintervals; in 

 the interior of each of them we have 



1 [y''' K {X, I, >l) - «/'■ K' (X, §, A) + y' K" (^, §, A) - ^ K" (.r, c, A)] =r 

 dx 



= y"" K{x,l,))-yK""{x,l,X). 



Integrating the equation over the subintervals, adding the results 

 and regarding that y" = y" = K" {x,g,X)= K'" {x,lè,>-}=0 forx = 

 and .1- = / and that y, y',y",y"', K ,K', K", K" are continuous every- 

 where except K"' in §, we find 



-ya)= f\ y" (*) K (.r, £, ;.) - y (x) K"' {X, i, A) \ dx. 







Replacing y"" by cf—T-y from (1) and K'" by — //T from (2) 

 we get 



y{S)=CK{x,i,X)^(x)dx 







or interchanging x and § and observing the symmetry of K {x, %, X) 

 with respect to x and £ 



y(x)=iK{x,g X)q;(§)dl 

 



If the beam is not loaded by q{x), but by N loads Q,, concen- 

 trated in the points §, , we have 



N 

 y{^v)= 2 Q'iK{x,èi,X) 

 1 = 1 



where Q'i = QijEI. If the beam bears both the load q {x) and the 



loads Qi we have 



/ 



y (x) = Ck (x, £, A) q' (£) dè+ 2^ Q'i K (x, ^„ A) . . . (4) 







From (4) it follows that y is a meromorphic function of A with 

 the poles 0, — (^P«//)* and — (^V)' 1'^''^ ''^ easily seen from sub- 



