24 Proceedings oj the Ro>jal Irish Academy. 



Fig. 4 furnishes a neat geometrical proof of the heights of the apexes 

 C, J). E, and F. Consider the quadrant A^Oi ; its base is 3 feet, being half the 

 distance of the wheel Wi = 12 tons from the centre of gravity of the locomotive. 

 The height of the segment is 9 foot-tons, being the square of its half base. Xow, 

 the vertical through An meets the tangent mn at a point twice as high above 

 the base. Hence mn slopes at an angle to the horizontal, whose tangent is 

 twice the square of the half base of the segment AjUi divided by its half base. 

 So that the tangent of the slope of EF to the horizontal is given numerically by 

 the distance of TT^, the 12-ton wheel from the centre of gravity of the locomotive. 

 To get the amount that E is higher than F it is only necessary to multiply 

 OFi by this tangent, when we get the product 12 x 6 = 72 foot-tons. In the 

 same way the height of i^ above BO is the tangent of the slope of FC, that is, the 

 distance of the wheel JF^ = 9 tons from the centre of gravity multiplied by 

 FiC when we have the product of 13 x 9 = 117 foot-tons. 



Then the maximum bending-moment at each point of the span is given by 

 the vertical height from the polygon to the parabola. The maximum of 

 maxima in any field — say, the fourth field — is to be found by producing EF to 

 meet the parabola at J and c, then he is to be bisected at the black spot where the 

 height to the parabola gives 261 foot-tons, the maximum of maxima ; provided 

 that the bisecting point falls, as it does, on the side EF, and all the wheels are 

 on the span. In the same way for the thii-d field, the height at the centre of 

 the chord of the parabola given by DE produced when measured vertically to 

 the parabola gives 256 foot-tons. For the other thi-ee sides of the polygon 

 produced to give chords of the parabola, the bisecting points do not fall on 

 the sides. Observe, too, that the black spot bisecting the chord cEFb faUs 

 3 feet, measured horizontally, to the light of the vertical through the middle of 

 the span, so that the graphical diagram, fig. 5, gives the maximum of maxima 

 for the 42-ton loco., crossing the 42-foot gii-der, to be 



iM^ = 261 foot- tons. 

 that is, when the foiuth or 12-ton wheel stands 3 feet to the right of the 

 middle point of the girder. 



The polygon ACBEFB, which is mechanically subtracted from the para- 

 bolic locus on fig. 5, is the bending-moment diagram for four fixed forces 

 acting upwards at the junctions of the fields ; 5 tons at C, 8 tons at D, 10 

 tons at E, and 7 tons at F, as these forces would give the moments 85, 145, 

 189, and 117 foot-tons of negative bending at those points. Generally, then, 

 the polygon is that due to a set of upward fictitious forces at the junctions of 

 the fields whose magnitudes are numerically the same as the distances between 

 the wheels multiplied by the ratio of the v:eiglit of the locomotive to the length 

 of the span. 



