3.0 ProceediugH of the Koifcd Irish Academy. 



fields where the last arc ended. On fig. 6 the vertical .scale is the same as 

 the horizontal scale because of the load and span being numerically equal. 

 The height of ai will scale 16'16 parts on the scale upon which the height of 

 the crown of the semicircle scales 21. Then 



Jl.% = (16-16)^ = 261 foot-tons. 



On fig. 7 is shown the 42-ton locomotive standing on the 42-foot girder. 

 On fig. 8 the span is divided into five fields proportional to the loads. 



Then the centres for the arcs are set off about the middle of the span at 

 Iwlf the distances at which the wheels stand from the centre of gravity 

 of the locomotive. 



The centre of gravity of the locomotive is defined by (?42 on fig. 7, where 

 the end links of the link-polygon meet, the link-polygon being drawn to a 

 polar distance of 10 feet. The five circular arcs are then drawn on fig. 8, each 

 beginning at the junction of a field where the last arc ended. The vertical 

 scale is made so that the height of the crown of the semicircle shall scale 21. 

 Next the crown of the circle in the fourth field is ruled over to the scale and 

 reads 16*16, and this when squared gives 



M-^ = 261 foot-tons. 



The front wheel, 5 tons, is now dropped off when G^^ fig. 8, defines the 

 centre of gravity of the remaining four wheels ; from G^3, a perpendicular is 

 dropped on the vertical from the ruling- wheel 12 tons; this perpendicular is 

 bisected, and the half scales 1*85 feet. On fig. 8 the new span, with the 5 feet 

 at the left end left off, is bisected at the ring, and the centre for the arc 

 corresponding to the wheel, 12 tons, is laid at 1-85 foot to the right of the ring, 

 and the centres for the other three arcs spaced relative to it. The arcs are 

 then drawn for the second, third, fourth, and fifth fields, and the crown of 

 the arc on the fourth field is ruled over to the scale where its height reads 

 14-64 ; and squaring this, we have if = 214 foot-tons, a maximum, at 1'85 feet 

 to the right of the centre of the 37-foot span when the 12-ton wheel stands 

 over it. 



Dropping off the second 5-ton wheel we have the centre of gravity of the 

 remaining three wheels defined at 6*3, ; and the perpendicular from this point 

 upon the vertical from the ruling-wheel, 12 tons, gives, when bisected, 

 0'734 feet. This distance is laid off to the right of the ring marking the 

 middle of the 32-foot span and the two remaining centres placed about it as 

 before. The three arcs are then drawn, the highest crown ruled over to the 

 scale, where it reads 13-04. Squaring this we get if =170 foot-tons, a 

 maximum, at a point 0-734 feet to the right of the centre of the 32-foot span 

 when the 12-ton wheel stands over it. 



