Alexander and Jackson — Polygons to Generate Diagrams, Sfc. 15 



The diagram generated above the base by B, the carried point, corresponds 

 to a loco, with loads on its five wheels thus — 



w(y3-3/2), 2w, 3w, 2w, wi^3-'ij2). 



They are spaced 2a apart, giving an extreme length 8a, which is more 

 than (ia + 2«J3) = 7'564rt, the span. With the carried point B a little 

 further out from the centre of the hexagon, the girder would be slightly over 

 8a in span, and would give the 8a-locoi space to move and bring its wheels 

 into their most trying positions without any wheel leaving the span. 



It will be seen that this is like the practical problem by comparing it 

 with fig. 10 ; only the shuttle loco, above with the weight mostly in the centre 

 and trifling at its ends is not usual. If its circumscribed circle replace the 

 hexagon, the carried point B describes a trochoid. 



Fig. 11. 



The part of the diagram below the base on fig. 11, if turned upside down, 

 corresponds to a girder with a uniform load double its span, together with 

 two downward loads 2a each at the two points where the arcs cross. See 

 fig. 7, which, like it, is produced by a polygon rolling inversely. 



To return to the upper diagram on fig. 11 produced by direct rolling 

 besides loco, just described, it also corresponds to a uniform load numerically 

 twice the span, with iqnvarcl loads numerically 2« each acting at the junctions 

 of t\\Q fields. See fig. 8. 



If the hexagon turn over inside a duodecagon with the same sides, the 

 arcs will only have half the areas shown on fig. 11, and their sum equal once 

 the area of the circumscribing circle. If it turn over outside the duodecagon, 

 the area will increase to three times that of the circle. In the limit, when both 

 polygons become circles, we have, adding the area of the rolling circle itself 



