253 



stitutiiig (3j in (4) and integrating term by terna, wliicli is permitted, 

 tlie series (3) being uniformly convergent. 



Expanding y in a series of" ascending powers of ^. (the first term 

 will in general contain A~i) the expansion will generally be con- 

 vergent for j>l| <^ C'^i//)* ; only if the term with the denominator 

 I -\- {'^I'ljiY cancels, the expansion will be valid for larger values of 

 ;.. In case the Qi's are zero this occurs if q {ai) be orthogonal to 

 (p^ (x). We thus see that if the expansion of paper 1 be exact and 

 if not by chance 



fi 



1 9 («) Vi ('^) d--*: = 







it converges only if 



kl* 



— < (2p,r = 500,54665 (5) 



M/J = 



From (4) we deduce a formula which will be of use further on. 

 Supposing the beam to bear only a load p (.v) pro unit of length 

 and to be in equilibrium, we will have 



I p (x) dx ^ \xp{x)dx^Q 







or which is the same 



\P (•'^) 'fo (*•) d.c = \p (.») ipo {x) dx = 0. 

 



Now from (4)^ in which (^ (.«) is to be replaced by p (.v)/EI and 

 in which Q'i = 0, we have 



y{x)=jK{x,è,X)pa)dS, 



where Kix,^,).) arises from K{x,S„k) by omitting in (3) the term 

 with the denominator k. Putting ^ = K{:v,i,X) changes into 



^ . ^^ V ^" W f" (^^ 1 V '^" (^'^ »''" (^) /«i 



A [x, i) = ^ \- ^ , . . . (0) 



^'' u = l (^PnjX ^n=X (2?„/,r 



and we get 



y(x) = ^jJK{,;i)p{^)dl ^^^ 







J 7* 



