Alkxanoek and Jackson — Polygons to Generate Diagrams, ^'c. 3 



The loads are expressed in half tons, so that the total load shall be 

 numerically equal to the span. The three parabolic loci are segments of the 

 same parabola, and from the above equality its modulus is unity ; that is, the 

 height of any point on the parabola above the base of a segment equals the 

 product of the two parts into which its base is divided by a perpendicular 

 dropped upon it. We can now replace these segments by semicircles, when 

 we have a locus of the square roots of maximum bending moments. The locus 

 near the ends completely changes as the wheels leave the span. For the 

 purposes of tliis paper, we assume that the girder overhangs its two supports, 

 and now the locus between the supports is complete. This is shown on 

 fig. 3. Here the two wheels equally share the load. 



Consider BA^L^ and CA.-,L^ to be semicircles replacing the parabolic 

 segments, and it is evident that the circular arcs BAJ>A.^C could be struck 

 out by D, the vertex of the triangle S^DS.^, lying first on its left side 6\Z>, 

 then turning over on its base, and again turning over on its right side DS,. 



This triangle SjDS.2 we call the " generating triangle" of the square roots 

 of the maximum bending moments on a girder of span equal to the perimeter, 

 the stress being due to a locomotive of a weight numerically equal to the 

 span, and equally divided on two wheels at a distance apart tivice the base of 



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