16 Proceedings of the Royal Irish Aea<lemy. 



insidcj twice the area of the rollmg circle, or half the.area of the larger circle, 

 but outside four times the area of the rolling circle. So that if a circle make 

 a complete exemsioii on a semieirde of double its radius, rolling inside as it 

 advances and outside on its return, the tracing'-point will draw a closed 

 curve with an area six times that of the rolling circle. One part will be straight 

 and equal to the diameter. It is evident that the area thus enclosed is the 

 same if the path be any plane curve, provided the curvature be nowhere 

 sharper than that of the rolling circle. 



For the diagram generated by A, a comer of the hexagon, the measure of 

 the downward uniform load is I'la, and four upward forces 2a each, at the 

 Junctions of the fields. In the eircumseribed circle, suppose the hexagon re- 

 placed by a 12-Eided polygon, there would be ten upward fore^, each measured 

 by twice the side of the new polygon. In the limit, when a circle was reached, 

 the sum of the equal upward forces would be the same as the downward 

 uniform load, but they would be spread sparsely at the centre of the girder, 

 and closely at the ends. 



Ths Cydoid. 



If we consider the inverse problem— namely, what is the manner of 

 loading a girder so that the diagram of the square roots of the bending 

 moments shall be the cycloid — ^we see that the total load is zero, that the 

 locus of the load is a curve, convex upwards, lying above the base for a 

 large central part of the base, erossii^ it near the ends, and reaching far 

 down at each end. The areas of the two parts below the base are the two 

 supports. 



Kow, as the areas for load, shearing, and bending for fixed loads are 

 derived by suee^sive int^ratiou, we propose to determine the shape of the 

 load by differentiation from the cycloid. 



"With a the radius of the rolling circle, we have 



x = a [Q - sin 0), y = a (1 - cos 6), 

 dx dii . r, d , ^ ^ . - 



d d' 2 '" - ?/) 



"With h to determine the scale, the equation of the load-locus is 



2{a-y) 



"J 



= iz 



y 



"With h = a, the equation takes the form (2a - rf)y = 2ob*, which leads to 

 the neat construction of the load-locus shown on fig. 12. The locus is seen, 



