254 



This represents tlie detlecliori of the beam under the conditions 

 that the beam be in equilibrium and that the ground be absent; 

 it is such that 



I y (x) d.v = la; y (x) dx = 



. (8) 



since K{.r, S) is orthogonal with respect to <f\{j') and ^\{x). Bj the 

 conditions (8) the deflection is perfectly determined and (7) repre- 

 sents it. 



5. We shall now prove that the series deduced from (4) agrees 

 for X =^ k' with the series of paper I. Representing the iterations 

 of K{x, £) bj K,{x, ï), K,{x, §).... we get for \1\ < (2/-.//)^ 



K(x, g, I) = K{.v ,§) - ; A-, {x, S) +;.' K, {x,S.) . . . , 

 Ck{x, è, X) q' (?) di =j K [x, I) 4 {I) dl - ïSk, (*, i) 4 {I) d| + . . . , 







as is proved in the theory of integral equations. From this it follows 

 that (4) for |A| <(2/'i//)' takes the form 



y {.'<') = y. («) + y. C^-) + »/. (•••) 



■ (9) 



where 



y» {•'^) = 







+ (P, (x) ^ Qi if, (§,) + y\,, (.f) 2 Qi V>. (£.) j 

 » = 1 1 = 1 I 



I 

 y, {X) = ^JK(X, g) q (I) d§ + ^^. J Ö. K(x, èi), 



y , (X) = - 't' I ^fr (■'' 5) 3 (') ^5 + ^ . f ^ <3. K, (.., § ) [ = 



= -^JKix,i)yAè)dè, 



