Ai.KXANDKR AND Jackson — Pobigoiis to Geiwrdte Diugruim^ cVc 10 



The intrinsic equatioiis of the cycloid, modified cycloid, and its evohite 



are , . . . (t> , -, . a 



s = -ka sin (/), s = 4« sin -~ , and s = la sm ~ ■ ■ 



In this way we have the cycloid with the generating circle sqneezed ont. 



The inset fignre shows the area of 4 circles inclosed by the cycloid and its 

 evolnte, with a joint perimeter of 8 diameters; and as modified into an 

 epicycloid with its evolute, inclosing the area of 2 circles, with a joint 

 perimeter of 6 diameters. 



This epicycloid can be generated either by the disc, fig. 13, having a 

 vertical bar that wraps round the evolute, or independently by a circle of 

 diameter two-thirds that of the disc rolling outside another circle of double 

 its own diameter. 



If we contrast the figs. 7 and 8 we see that as the direct rolling polygon 

 of the one becomes regular, so also does the concave base of the inverse 

 rolling polygon on the other. When the sides increase indefinitely and the 

 one polygon becomes a circle, and the locus ABCDEF a cycloid, then the 

 other polygon becomes equilatercd, its two long sides equal to the span AF 

 and its concave base as well. In form the base is like a parabola. The limit 

 of the locus ABCDEF, fig. 7, is like half of an ellipse, its area being the same. 

 The semi-axes are na equal to half of AF, and the height of I) equal to the 

 sq. root of 'Itt-ci- - -icr, derived from the relationship of the depth and heiglit 

 of D, below and above the semicircle on figs. 7 and 8, respectively, and shown 

 also on figs. 5 and 6. 



i'rom this relationship among the cycloid, semicircle, and limiting value 

 of the locus ABCBEF on fig. 7, all on the same base AF, the origin being 

 at A, with X = a{d - sin 0), we find 



-V = 2^(1:- - 6) - 4(77 - 0) sin 6 + (1 + cos-6) - 4. 



Conjugate Load Areas. 



In a former paper read to the Eoyal Irish Academy* on two-nosed 

 catenaries, we developed their application to the design of segmental 

 masonry arches. It was adopted later by Professor Howe in the chapters 

 of his elegant treatisef that dealt with masonry arches in general. To meet 



* " On Two-Nosed Catenaries and their Application to the Design of Segmental 

 Arches," by T. Alexander, c.e.. Professor of Engineering, Trinity College, Diililin ; and 

 A. W. Thomson, B.sc, Lecturer in the Glasgow and West of Scotland Technical College. 

 Trans, of the R.I. A., vol. xxix, part iii, 1888. 



t " A Treatise on Arches," by Malverd A. Howe, C.B., Professor of Civil Engineering, 

 Rose Polytechnic Institute, Terre Haute, Indiana. New York : John Wiley & Son. 

 London : Chapman <fc Hall, Ltd., L897. 



[2*J 



