1879.] On Most General Problems in Continuous Beams. 495 



And also, if we know how EI alters, we can find the following sums, 

 each of which is denoted by a symbol : — 



dx 



oEI 



xdx 



=d 



and also 



J x mdx 



d . dx=n 



Jo 



Jo EI 1 



pi xdx_, 



Jo Er _/ ' 



JoBT =?1 , 



(3), 



r 



>.<&= 21 



(4). 



It is very easy to write out the values of these summations in 

 certain simple cases, and the calculation in the most general cases 

 can always be made with a very fair approximation to accuracy, and 

 with not much risk of mistakes being made, by the graphic methods 

 described below. 



The bending moment at any point P is — 



M=M — SqX+wi 



(5) s 



but at Q we have 

 and 



and hence 

 therefore 



x: 



M=M 



So that expression (5) becomes 



(6). 



(7).- 



Substituting this value for M in the differential equation (1), and 

 integrating once, remembering that when x equals nought, d, f, and g 

 vanish, we see that if a is the inclination (very small) of the beam 

 downwards from the horizontal at 0, and if a is the inclination of the 

 beam at any point P, then 



dy 



dx 



or tan « or «=« I M. d — S / + g 



(8). 



