255 



y„+i (^) = - .^JK{.v, §) y„ (ï) d§, 



EI. 



Each of tlie functions //„(.i), except y,{x), satisties (8). We shall 

 now prove the terms J/o' !Jv !h' ■ ■ ■ to be the same as the corresponding 

 quantities of I, 5, from which it will follow that the series 

 y» + ^1 ~f" • • • • agrees with the series of 1,4. Indeed in the first 

 place yo(.r) is a linear Kinction of .r; the function ky,(x) represents 

 the linear load «;i' + (J, which is defined in I, 5 and is statically 

 equivalent to the given load. For we have 



/ I 



^ ƒ//, i^) <P, (■^) dx =J<f. iS)q{ê) '/I + J Qi cp, m, 







; ; 



kjy, (*) rp, (*) d.v = Jxp, {^)q(i) d + ^^ Q, xp, (g,). 







or substituting in it the expressions found for the functions q>, (x) 

 and tf», (x) 



I I 



Cky, (w) dx =Jq (§) d^ + ^ Qi 



~ 



/ I 



ƒ (^-i) ky, (x) dx =ƒ(§-!) ^(§) dg + 2^ Q, (§.— è), 







which proves the proposition. 



Omitting from f9) the deflexion y„ the remaining terms represent 

 the remaining deflexion. This becomes ?/, for ^ = and so y, 

 represents the deflexion which the beam, if not supported bj the 

 ground, gets under the influence of tiie load that remains after 

 subtraction of m x -\- ^ from the given load. As besides //,(.(■) satisfies 

 (8), it is identical with the quantity y, of I, 5. 



The reaction of the ground, arising from the deflexion 7/1, represents 



a load — ■ ky^ of the beam; by this load the beam, if not supported 



by the ground, would get a deflexion, which we may calculate 



from (7) viz. 



I 



EIJ 



K{x,l)y,(^)d^. 



