6 Proceedings of the Royal Irish Academy. 



twice the span, together with a concentrated load at the centre of 20 tons, 

 numerically twice the base of the triangle. 



The shearing force locus begins at the left support with 36 + 10 tons, falls 

 at the rate of two tons vertical to one foot horizontal, and would pass 

 through zero at 5 feet iMst the centre, so that here is the centre of the 

 circular locus ; but it falls suddenly 20 tons at the centre of the span. 



On fig. 6 the shearing force locus begins with 36 - 10 tons at the left 

 end, crosses the base at 5 feet before reaching the centre of the span, 

 continues to fall, but suddenly passes up through the centre, due to the 20 tons 

 iqnoards there. It again crosses the base at 5 feet beyond the centre. For 

 every crossing there is a corresponding max. or min. of bending. In 

 this case a min. at the centre and two equal max. of maxima at 5 feet, half 

 the base of the triangle, on each side of the centre of the girder. These 

 super-mas., 31^ = i\f_- = 169 foot-tons, are squares of 13, the sides of the 

 generating triangle, and are the radii of the two circular arcs, greater than 

 quadrants, meeting at D over the centre at a height less by 5^ = 25 foot- 

 tons than the max., giving 144 foot-tons. 



The dotted semicircle standing on the base of each fig. 5 and fig. 6 gives 

 square roots of the B. il. for the uniform load alone, the height at the centre 

 being 18^ = 324 foot-tons, an average of the values for 1> and I) on the two 

 figures. 



The case, fig. 6, for direct rolling is of importance because it also is the 

 bending stress diagram for a 36-ton locomotive, with the load equally 

 divided between two wheels spaced 20 feet apart. 



The Irregular Generating Polygon. 



Two reciprocal generating polygons are shown on figs. 7 and 8. 

 Each has begun to roll clockwise, the vertex A ha-\dng traced the first part of 

 the arc AB. The base of one polygon is a co7ivex polygon 2J, 4, 5, 3J, and the 

 base of the other is a concave polygon 3J, 5, 4, 2|. lu each case the girder is 

 42 feet in span ; the uniform load 84 tons, or numerically double the span. 

 There are also four concentrated loads of 3, 8, 10, and 7 tons dividing the 

 span of the girder into five parts of 5, 5, 11, 12, 9 feet. On one figure these 

 four loads are downward, an every-day thing, but on the other they are 

 lopivard — an arrangement leading to a locomotive equivalent to the uniform 

 load, numerically twice the span, together with the four upward forces. The 

 magnitudes of these four forces are numerically the four intervals between the 

 five wheels of the loco., while the five intervals on the girder are numerically 

 the same as the weights of the wheels. 



