10(5 
Mlv\ h. N. (5. Kll.ON (>N AN ArPKOXl.MA’l'K SOIATION FOK OKNIONC A 
'I'lu' oxprosslous (80) ami (00) show us that in this case the stresses ami the 
aisplaeements are ohtainecl as the cliiVereuce of two functions taken with the 
('Ntremities of the layer as orioius. 'fhe series are everywhere uniformly ami 
ahsolutely eonvergent inside the common part t'f two circles ot radius 4h described 
about each of these extremities as centre. It follows that if these series are to he 
valid anywhere, the length of the layer must m>t exceed 8/>. Ami if they are to he 
valid round each extremity the length of the layer must be less than ib. If these 
conditions he not tultilleil, then we have to fall hack on the results (/4), (/a), (/b), 
ami (8;A for \\ Q, S, U, Integrating these we obtain formula' valid over the 
whole beam, and these again may he expanded in powers about any point we please, 
as has been previinisly shown. The results are rather long and do not seem to 
present siitlicient interest to justity the writing out ot them at length. 
Assuming -<r' < 4/>. so that the expressions (8i>) and (80) are valid over a. region 
enclosing the laver of application of the load, we see that here no displacement is 
either intinite or discontinuous. hor in the limit, both ((/' + .!•) log )\ and // log r, 
are /.ero when .r = — a\ tf = 0 ; and in like manner (.r — «') log r., and // log r.. are 
zero V hen .r — , y = 0. 
' 
M’ 
' 
- — ^ 
Fla Tic (>f 
SUTTlTTlCtrU i/ = o. 
- — 
■ 
B 
.w 
Fig. iii. 
The shear 8' is continuous. 
’fhe stresses T', Q' however are discoutiuuous at the extremities ot the layer. Ibis 
indeed is obvious in the case of Q', since it is one of the data of the problem. But it 
is curious to note that B' is discontinuous at those points by precisely the same 
auuniiit as Q'. 
^ '28. Cv'sv(*bn ln\im unJcr Ttro Equal and Loa<ls, or hinting uqh'ui a Ingid 
Smooth Plano. 
If we take the solution we have obtained, turn it upside down, as it were, and 
superpose it tv' itself, we v'btaiu the solution ot the prv'bleiu v't an intiniteK kuig beam 
