18 Proceedings of the Royal Irish Academ/j. 



10-foot space, and is 6 feet from the 12-toii wheel and 4 feet from the 11-ton 

 wheel. 



The locomotive is shown both on tig. 1 and fig. 2 standing in the same 

 particular position. On fig. 1, the positions of the wheels are given left and 

 right of 0, the centre of the span. An instantaneous bending-moment 

 diagram ABCDEF is drawn to scale for that particular position of the 

 locomotive. 



On the other hand, fig. 2 gives the position of the wheels left and right of 

 G, the centre of gravity of the load. By inspecting these two figures it will 

 be seen that the wheel TF'4 = 12 tons, which we will call the ruling-irhed, since 

 under it, as will be shown subsequently, the greatest bending-moment occurs* 

 lies hi = 3 feet to the right of the middle of the span, while G lies 3 feet to its 

 left. On fig. 2 the bending-moment under the ruling-wheel is given to scale 

 by Si Ai, so that At is identical with the apex D on fig. 1 . We now contem- 

 plate moving the locomotive left and right of the position on the figures, so as 

 to find the locus of the apex D on fig. 1. It is well at this stage to conceive the 

 girder as extending some distance beyond each support as shown on the model, 

 fig. 10, at the end of this paper. 



On fig. 2 the locomotive stands with the nding-ichcd at a distance 

 (21 - 3) = 18 feet from the right abutment, while the centre of gra^aty G is 

 18 feet from the left abutment. Since the load and span have the common 

 value of 42, therefore the push-up at the right abutment is 18 tons, because G is 

 18 feet from the other abutment. The bending-moment imder the rvliag- 

 wJieel is 18 tons multiplied by 18 feet less 63 foot-tons, the product of the 

 next wheel load 9 tons by its distance 7 feet from the ruling-wheel. The 

 height of B, fig. 1, and of At, fig. 2, is 



tons. ft. 



4J/_3 = IS X 18 - 63 = 261 foot-tons. 

 The subscript figures meaning that the fourth wheel stands 3 feet to the 

 right of the middle point of the span. If the locomotive moves one foot 

 either to the left or right, one of the " eighteens " in the above product 

 becomes seventeen, and the other nineteen, and 



tons. ft. tons. ft. 



J/-2 = 19 X 17 - 63 or J/.4 = 17 x 19 - 63. 

 These are the common heiglit of the locus of D, fig. 1, at one foot left and 

 right of its figured position according as the r!<;i«^-wheel JF4 arrives at the 

 one or other point. Hence A^, fig. 2, is the vertex of a parabolic right segment, 

 its half-base being 18, and its height, IS x 18 : but placed with its vertical axis 

 3 feet to the right of the middle of the span and with its base lowered 

 63 units below £C, the base of the bending-moment diagram. It will be 



