26 Proceedings of the Poyal trish Academy. 



(50, 40, or 30 p. c.) a percentage for granite, sandstone, or brick riugs. As 

 these voids alter the true shape of the spandril-area, it might be taken as 

 having the parabolic shape EAyB, which is the north-south load on the 



AS-^, quadrant of fig. 2a, with a parallelogram am added of depth - 



acting towards the rib, and raises the thrust at £, fig. 3a, the springing, to 



^, doubling its value. A like parallelogram </«, fig. 2a, acting towards the rib, 



is added there to the horizontal load-area, and now the thrust at the 

 crown is nil. This value is verified, as the normal load at the crown is now 

 zero. For the parabolic spandril-load on the quadrant AS' we have the 

 horizontal conjugate load-area, shaped figure-eight-wise, mapped out by 

 the two boundaries — a parabolic quadrant chk and a vertical line I)h7i, 

 the lower part acting towards the rib and the upper away from it, their 



2r^ 

 common area being Q,' = Q/ = •1287-', or - — — . It is convenient on fig. 3a 



y \/ o 



to replace that vertical boundary by an at 45° to the horizon, when the other 

 boundary chk becomes a right parabolic segment. They cross at Ji on the 



same level -— ^ as before, and the point a on the rib is Eankine's point of 



rupture ; and the limiting position of 6 is <p, given by 



COS^ = "V" = '^11- 



If a parallelogram DU for the superstructure be added to the load, 



r* 

 say, AD or au = a sixth of the radius = tt, it is at once added also to the 



o 



conjugate load-area by the draughtsman placing his set-square at the point 

 u, and drawing the one-to-one dotted boundary uo to supersede an. The 



,.3 5.3 



thrust at the ci'own is now Q^ - Q2 = 7',, = ^, and at the springing J, = — ' 



Part of the added parallelogram may allow for the excess density of the 

 masonry ring. The boundaries of the new figure-eight-like area cross at 



p, so that |3 is now the point of rupture, and cos 6 = ^ = •707. The 



solid backing required by the arch §, = phko = - — ,> and the leaving 



out of Q-i, the negative part of the backing, makes T„ increase by 



r- ,.! 



g- (\/2 - 1); and since the normal load at A, tlie crown, remains ^, the 



radius of curvature of the modified line of stress must be greater than r in 

 the ratio -^2 : 1. The modified curve from ^ to /3 is a modified catenary, 

 flatter at A than the circle, and sharper in curvature at about 30° near /j. 



