36 Proceedings of the Ro>/al Irish Academy, 



foot-nm on a girder of 42-feet span. The effect on the scala would, however, 

 be different : in the ease of the increase of span the scales both of length and 

 bending-moments would be made finer iu the ratio of 42 to 56, or 3 to 4, 

 while in the case of the addition' of the iiniform load the length-scale would 

 be unchanged, while the load and bending-moment scales are made finer in the 

 above ratio. 



It is evident that the diagi'am of maximum bending-moments for the 30-foot 

 42-ton locomotive of fig. 5 rxmning on a 42-foot span which already carries 

 a uniformly distributed load of 28 tons and drawn to scales of, say, 

 3 feet, 4 tons, and 48 foot-tons to the inch is identical in form and size 

 with the diagram for the same locomotive crossing a span of 56 feet, and 

 drawn to scales of 4 feet, 3 tons, and 48 foot-tons to the inch. Now, on changing 

 the scales of the latter figure to 3 feet and 4 tons to the inch, it is seen to 

 represent equally well the diagram of maximum bending-moment for a 22|-foot 

 o6-ton locomotive crossing a 42-foot span. So that the effect on the diagram 

 of maximum bending-moments of the addition of a uniform load is the same 

 as that produced by shorteniwj the intervals between the wheels in the ratio 

 of the loco, weight to the sum of the loco, weight and half the uniform 

 load, and making the several wheel loads heo.vicr in the reciprocal ratio. This 

 is illustrated in fig. 9, where the derivation of the diagram of square roots of 

 maximimi bending-moments for the loco, of figs. 1-5 is effected when the 

 loco, is supposed to cross a bridge of span 42 feet already carrying a uniform 

 dead load of two-thirds of a ton per foot. 



The method is useful as illustrating clearly the efi'ect on the bending- 

 moment diagram of the addition of a iiniform load; but it requires to be 

 applied with some degree of caution, as it must be remembered that the change 

 in the character of the diagram which takes place when a wheel goes off the 

 span occurs when a wheel of the real locomotive goes off, and as the ideal 

 locomotive is shorter than the real, it might readily be forgotten in examining 

 a particular portion of the locomotive that one wheel was off, since the 

 corresponding wheel of the ideal locomotive might be well within the span. 



KinemoMco.l Model, deraonstraMTig the variations in hendvng-rtuyment at all 'points 

 of Oj girder-bridge as a locomotive comes across the bridge. 



This model is specially designed to exhibit the manner ia which the 

 diagram of maximum bending-moments, consisting of arcs of parabolas (as 

 shown in fig. 2), is traced out by the vertices of the link-polygon corre- 

 sponding to the loads on the wheels of the locomotive, as the locomotive 

 moves across the spam The model (see fig. 10) consists of two parallel plates of 

 mahogany each 18i in. x loin., the front or face-plate being f in. and the back 



