241 

 and to put 



,r = z, {.V) + i- /, (x) + .f' X. (^) + • 



This function satisfies fonnall}' tiie equation 



cp (x) + i-' l"^-^— '/' (*) f/^' = g' + C .^ + i> 







and the expression y, which follows from it: 



q - <f <i - (q +r,x +i>.) - A' •/: (^) - k" X, (.^) - 



y — 



k' 



C,x ^ D, 



k' 



X, (*•) - 't' X, (.^0 - ^'* •/, (■''•) • . 



satisfies foi'inally the conditions, imposed at the ends x =:0 and x = I. 

 For, substitnling the expression </> in the integral equation we 

 obtain — provided that it be allowed to integrate term by term the 

 series, which occurs under the sign of integration: 



C,x-{- D,-k' ((7, X f D,) - k" (C, X + I),) — .... = Cx ^D. 



If the series, which appears in the first member of this equation, 

 converges, there can be disposed of the constants C and D in such 

 a manner, that the e((uation becomes an identity. 



Of course it would now be necessary to examine the convergency 

 of the described process of iteration. 



For this investigation however we refer to the paper of iVIr. J. 

 Dkoste, which follows this. We will state here only, that conver- 



gency is sure, ii j^rj <i ^^^, «-"d go on to demonstrate in which 



manner the process can be graphically performed. 



5. At the first place the system of forces, which loads the beam, 

 is substituted by another load, changing linearly, (g„ = o.r -)- ^), and 

 which i.s staticallj' equivalent with the first. 



This substitute load causes a sinking down of the beam, determi- 

 nated by 



« X + /? 



This y, can be considered as the first approximation of the 

 required ?/, and can be brought in relation with the expression 

 C^x -\- Z),, which is defined in N°. 3. 



Indeed, « .c -(- ;- satisfies the equations 



