Reactions at the Points of Support of Continuous Beams. 271 



and so on, there being as many equations as intermediate points of 

 support. 



If the function m/I is assumed to represent the intensity of a new 

 loading on the girder, it can easily be shown that the expression 



o 



represents the value of the bending moment at the point A 1} due to 

 this new loading, considering the beam as supported only at each 

 end. 



Let this expression for the bending moment at A x be called N l9 and 



that at A 2 N 2 , and similarly for A 3 



Again, we may write the expression 



ETo'Zi— J| ~ dx 2 j equal to Ri^/, 



»/ being the bending moment at A 1? due to a new " m/I " loading 

 obtained by assuming a unit force to act at A^ 



Similarly ^ETo"^- jj dx^j = B>,n L ". 



Hence the equations become 



IS^ = R 1 tc 1 ' + R 2 V+R 3 V + , 



N 2 = R^' +R, 2 n 2 " + R 3 n 2 "' + . . . . , 



N 3 = R 1 < + R/?7 3 ' / + iW'+ , 



&c, &c, , 



from which R l5 R 2 , R 3 . . . . may be easily obtained when the con- 

 stants have been determined. 



Lord Rayleigh has shown, in his ' Theory of Sound ' (vol. 1, p. 69), 

 that when a beam is loaded with a concentrated load at any point P b 

 and the load is transferred to a second point P 2 , then the deflection 

 at P 2 when the load is at P x is equal to the deflection at Pi when the 

 load is at P 2 , hence 



n 2 ' = n" ; — ; nj" = nl' ; &c, &c, 



thus reducing the number of constants to be found, or affording a 

 check on the accuracy of the working. 

 In the example appended 



