Alexander and Jackson — Polygons to Generate Diagrams, &,-c. 17 



by inspection, to cross the base when Q = 90° y = a\ and so di\ide the 

 half-base into segments, the arc of a quadrant pins and minus a radius 

 respectively 



tan di = --;- = — sm \). 

 ax if 



And the locus crosses the base at a slope of 2 to 1, corresponding to the 

 1 to 1 slope of the cycloid. Further, the areas below the semi-base and 

 above it, the limits being 



= (l 



1 1 a and 



TTft. 



are 



■)dx = - 2fl- 



cos a . dd 



dx = -2a^ cos e.d9 = 2a"- 



CYCLOID A DIAGRAM 



OF- 



u- — "^ 



THE SQ. ROOTS OF ^ 



/ 

 BENDING MOMENTS. 



' ^ 



Uf' 





l./- 



^7 



f 2«-^ 



,1 







\ 1 



^f 





\ 



'< 



:v^ 



^^ 



a(f-d. 



SUPPORT. 



P. 



-TvTS??) 



enfs 





A^ 



a 



/>C> 



ot 



ioci< 



/ 



o 



J-OAD^ 



%^!l'of 7? C//iD£R- 



a($^/) 



A!exanil«r aoJ Jackioo 



Fig. 12. 



A girder loaded in this manner over a middle nine-elexeiiihs of its span and 

 supported on the end segments would have the cycloid as its diagram of the 

 square roots of the bending-moments. 



E.I.A. PROC, VOL. XXXVI, SECT. A. f^] 



