14 Proceedings of the Royal Irish Academy. 



diagram on a large scale and compounding the triangles by lineal 

 measurements alone. 



Using a third scale, the upper inset would yield results for the real loco, 

 on spans between 42 and 56, including dead loads less than 28 tons. This, 

 with other positions of B, enables the template, fig. 10, to generate diagrams 

 giving general results for various spans upwards of 42 feet, and to include 

 any desired dead loads, but not for spans much smaller, as the hyperbola 

 would ciu've rapidly. 



A large wooden model polygon turning on a lath pinned to the black- 

 board and guided by a st^el strap jointed at 6, c, d, e, and /, and having a 

 chalk holder that can pull out from A to B, is made for demonstrating 

 purposes by Bixmi and SemjpfiJistall, Dublin ; also a celluloid polygon. 



The B/jjxdar Polygon. 



On fig. 1 are shown the diagrams generated by the regular polygons of 

 two sides (base zero), three, foiir, and five sides, with the corresponding 

 locos. The super-max. is given numerically by the radius of the arc nearest 

 the centre, and occurs either at the centre or distant half a side from it, 

 according as the number of sides of the polygon is even or odd. 



That the area of the circular sectors is twice that of the circle circum- 

 scribing the polygon follows from the theorem. If from one of a number of 

 points dividing a circle into n equal arcs chords be drawn to the others, the 

 sum of the squares of the n - 1 chords equals the area of the circle multipHed 

 by 2m : TT. 



As the two isosceles triangles, one with the base zero and the other with 

 the base equal to the sides, both sweep out the area indicated, it follows 

 that some isosceles triangle lying between them will sweep out the minimum 

 area lying between those two equal maxima. It is readily proved by 

 differentiating that 29 the vertical angle is given by 26 + cot 6 = -. 



On fig. 11 are shown the diagrams swept out, one by the comer A of the 

 regular hexagon, and two, above and below the base, by a point B carried by 

 the hexagon at a distance twice its side from the centre out through A. 



For the hexagon the corresponding loco, has its five wheels spaced 

 2a apart, and their weights are as follows, being the same as the fields into 

 which the span is divided, but having a replaced by w : — 



vj 3iv „ 3vj vj 



— — Ivj — 



2' 2' ■ 2' 2" 



The radii of the arcs covering the fields are «, a v/3, 2«, a ,/3, a. 



