OKA P. V.] MOMENT OF ROTATION PARALLEL FORCES. 47 



and hence that the line a Ic must be a tangent. Thus the moment at any 

 point is 



&y = % p lx $p a? 

 p being the load per unit of length, I the length, and the reaction at sup- 



port therefore ^-. Hence y = ~- (Ix ") for origin 0. 



a a H 



When the origin is at d, representing horizontal distances by y' and ver- 



tical by x, we have x = - y', and y = h a:', A being the ordinate at 

 



middle = 

 Hence by substitution 



or reducing 



', 2H , 



y -j-* 



which is the equation of a parabola having its vertex at d.~\ 



We may of course take the pole anywhere, and hence H may 

 have any value. It is in general advantageous in such cases 



(i.e., for uniform load) to take H = *=- We have then 



a 



f = Ix, 



and for y ^ or for the middle ordinate. we have x = -r' 



To draw the moment curve we have then simply to lay off 

 the middle ordinate equal to -Jth the span. The curve can then 

 be constructed in the customary way for a parabola. Any 



ordkiate to this curve multiplied by H = ^=r will then give the 



moment at that point. 



Enough has probably now been said to illustrate the applica- 

 tion of our method to the determination of the moment of rota- 

 tion, bending moment, or moment of rupture. The reader 

 will have no difficulty in applying the above principles to any 

 practical case that may occur. 



It will be observed that the customary curve of moments in 

 the graphic methods at present in general use, comes out as 

 a particular case of the equilibrium polygon for uniform 

 load. 



This polygon has other interesting properties, which we shall 

 notice hereafter. For instance, just as its ordinates [Fig. 30] 

 are proportional to the bending moments or moment of rotation, 



