92 SIMPLE OERDEE8. [CHAP. VIL 



the unit of length (when the pole distance is a force units), or, 

 what is the same thing, one unit of length is equal to a moment 

 units. The same equilibrium polygon can be used for any 

 number of girders of various spans, hejice the method is of very 

 rapid application. 



75. Absolute Maximum of Moments. Since for any cross- 

 section M is a maximum when a load lies at that section, a load 

 must also lie upon the cross-section for which M is an absolute 

 maximum. 



If the line A B slides upon the equilibrium polygon, altering 

 its length so that its horizontal projection is constant and equal 

 to I, it will envelop a portion of a parabola so long as its ends 

 move in the same sides of the polygon. [PI. 13, Fig. 46.] The 

 curve thus produced is therefore composed of portions of a 

 parabola. Let the ordinate D C correspond to the moment at 

 the point of application of the load P. DC will be evidently 

 greatest when A B is tangent to the curve at C, so that the 

 maximum of the moments occurring at D is given by the dis- 

 tance C D between the polygon and curve enveloped by A B. 



Let the prolongation of the sides upon which A B slides meet 

 in E, and F G be the tangent to the parabola at the point H in 

 the vertical through E, so that F H = H G, and let I be the in- 

 tersection of A B and F G. Draw through A a parallel to E B, 

 intersecting F G in K. Then the horizontal projections of A F 

 and A K are equal, since those of E F and E G are equal. 



Since, however, the projections of F G and A B as also of 

 A F and G B are equal, A K must be equal to G B. Hence 

 A I = B I. In a parabola the distances of the three diameters 

 passing through two points and the point of intersection of the 

 corresponding tangents are equal, hence the projections of H I 

 and C I are equal. 



The middle point I of the tangent A B lies, then, half way 

 between the angle D vertically below the point of tangency and 

 the intersection E of the sides upon which it slides. 



Since the projection of A B is I, its construction is easy. 

 The construction must, of course, be repeated for each angle, in 

 order to determine that for which M is an absolute maximum. 



The above principle may, then, be thus expressed : The mo- 

 ment at any load is a maximum, when this load and the result- 

 ant of all the loads are equally distant from the centre of the 

 girder. (See also Art. 48.) 



