2 Proceedings of the Royal Irish Academy. 



and the pentagon four times, the tracing point describing four ares. The 

 least polygon is the limitmg isosceles triangle when the base is zero ; its 

 vertex describes a semicircle. The base of each diagram so generated is the 

 perimeter of the polygon. From the one extreme of the limiting triangle 

 with two sides we may proceed to the other extreme of a polygon with an 

 infinite number of sides, when the tracing point will describe the cycloid. 



In all these diagrams the sum of the areas of the circular sectors is 

 iwiee the area of the circle circumscribing the generating polygon. The 

 triangular parts, shaded on the figure, are the polygon itself split up and 

 buried ia the diagram. This makes the total area of the cycloid three times 

 that of the generaring circle. See hexagon, %. 11. 



Load or: eac^^iyheel eqiLalTb its'Fj E ld' 



}'fheels J-paced 



a * oT 



Fig. 1. 



These diagrams give maxima stresses for certain dead loads as well as 

 moving loads. The arrangements of the loads, however, are not practical, 

 and so we pass on from these to irregular polygons, but will return to them 

 later on, as there are interesting geometrical traits connected with them. 



The Gffimrating Triangle. 



The fig. 2 shows a locomotive, the wheels 12 feet apart, and transmitting 

 24 and 12 half tons respectively to the 36-foot girder as it passes across. 

 The double parabolic locus gives the max. bending moment at each point as 

 the corresponding wheel comes over it. The larger wheel dominates two- 

 thirds of the span, as its weight is two-thirds of the load. The dotted central 

 paiaboUc segment standing on the span is the locns of maxima for the 

 adjacent girder, upon which the whole 36 half tons is concentarated by one 

 wheel, seen peeping over the loco. It is situated between the other pair of 

 wheels opposite their centre of gravity, which is 4 feet from the one and 

 8 feet from the other. 



