Mathematics. — "Ati application of tlie theory of integral equations 

 on the detenniruition of the elastic curve of a beam, elastically 

 supported on its lohole leuf/th". By Dr. J. Droste. (Communi- 

 cated by Prof. J. C. Kluijver). 



(Communicated at the meeting of March 24, 1923). 



1. Under the same title and at the same time a paper ^) of Mr. 

 BiEzENo appears in these Proceedings. The question, suggested, in N°. 4 

 of that (taper as to the validity of tiie process of iteration used in 

 it, will be answered here. 



For that purpose we observe that the function of x, satisfying the 

 differential equation 



-^^ + rv = 9'W (1) 



ax* 

 and the conditions at the ends of the interval, is a meromorpliic 

 function of ^. We might find it by means of the method of the 

 variation of constants and then expand it in ascending powers of >!; 

 the radius of convergence R of the power series that stands after 

 the first term (containing ^~i as a factor) might easily be calculated 

 then. After this it will be necessary to investigate wether it agrees 

 or not for X :^ k' with the series of paper I; it is only in the first 

 case that the latter series will be valid for k' <^ R. For the sake of 

 this investigation, however, and also in order to get an idea of the 

 proportionality of the functions t„(x) (vid. I, 7), we prefer to use 

 the method based upon the theory of the integral equation of 

 Fhedhoi.m. 



2. Wo construct a function of x, satisfying in the interval (0,/) 

 both the equation 



d*y 



5^ + '' = » .<'> 



and the conditions y" :^ y'" ^ at the ends, and being continuous 

 as well as its first three derivates everywhere in (0,/) with the only 

 exception of a saltus of the third derivate at the point ^: 



d'y 



</.v 



f + 



= 0. 



f-o 



') Referred to in the sequel as "paper 1". 



