Ai.KXANDKR AND Jackson — Pol/igons to Generate Diagrams, Sfc. 13 



the base, with hA produced each way. The steel point of the dividers will 

 go througli h, and the pencil point through A, and the first arc drawn till 

 he, produced both ways, comes on the base, when the steel point shifts to c. 

 In this way the five arcs are drawn. For good definition of the junction of 

 the ares, each arc should be drawn a bit both back and forward. A pin and 

 a pencil, or syphon ink-pen, may be used instead of the dividers. The 

 tracing point A will generate the smaller diagram, inset on fig. 10. See also 

 fig. 8. The two super-maxima, equal to the two central radii, will scale 

 (16)2 and (16-1)2, giving M= 256 ft.-tons and M = 261 ft.-tons, at 2 feet left 

 and .3 feet right of the centre of the girder. The scale for feet and verticals 

 is 4 feet to an inch. The arcs will cross on verticals, dividing the span 

 into fields numerically equal to the weights of the wheels. 



Another tracing point B is plotted by arcs from h and /of radii 19| and 

 21-2". It generates the larger inset on fig. 10, almost mathematically the 

 corresponding diagram for a girder 56 feet in span. The angles should have 

 been slightly changed, but as B is sensibly on the dotted vertical through A, 

 the two super-maxima dB, eB will scale with great accuracy. The crossings 

 of the other arcs will not lie so closely on the junctions of their fields. The 

 locus of the points such as B, which should be the dotted vertical line, will, 

 when plotted as above, be a hyperbola with foci at h and /, as hB and fB 

 constantly differ by two feet. The part AB will be sensibly straight. 



By using a scale of 3 feet to an inch, that is, coarser in the ratio 

 56 to 42, the higher inset on fig. 10 returns to the initial span of 42 feet, 

 while the loco, is replaced by the heavier, shoi'ter ideal loco, described at 

 fig. 9. The super-max., now '2\ to the right of the centre of the girder, 

 will scale 17*44. Squaring this, we have 304-31 ft.-li tons, as the unit of 

 load is now 1\ tons to make the load and span equal numerically. So we 

 have the position 2\ feet, and the value 405| foot-tons of the super-max. on 

 the 42-foot span for the 42-ton loco, together with a dead load of 28 tons. 



At the junction of two " fields " the bending-moment remains constant 

 as long as the pair of commanding wheels straddle across it, or one of them 

 arrives at it. Their values on fig. 8 are 100, 175, 256, and 180 foot-tons. 

 If we complete the circle of which CD is an arc, complete the chords CC\ DD', 

 and the diameter BC, then the tangent of CG'D is 11 divided by 13-2 + 16, 

 and we have the half angle subtended at the centre of the chord CI). This is 

 the extern angle at the point d on the polygon (fig. 10), and for the others: — 



cos J = -6, tan — = -215, tan — = -37, tan -— = -41, cos/= -4. 



^ A Ji 



The template (fig. 10) is quite accurately constructed by plotting the 



