Alexander and Jackson — Polygons to Generate Diagrams, &i-c. 29 



of the depth r, then gl batters at 1 in 2 for the next four-ninths of the depth, 

 and //.: at 1 to 1 for the last ninth. This boundary encloses the exact area 

 Jr^ just as the curve does. And, further, the triangular area and also includes 

 the area ^r', and acts towards the rib to resist the spreading due to the 

 weight of both the spandril and the collar ; see the triangle with a dotted 

 hypotenuse on fig. 4«. Removing the collar aivay requires that and 

 be removed away from acting on the rib. The two figures and and ahghl 

 standing on the same base ad, and the same side of it, the one, when thus 

 substracted from the other, leaves the figure-eight-like area abghn of alge- 

 braical area zero to resist the weight of the spandril-area alone. The area is 

 mapped out by the 1-to-l boundary an and ] -to-4, l-to-2, 1-to-l treble-batcer 

 boundary hglk. This figure is readily drawn on squared faper, represented by 

 dots on fig. 5a, all the points h, g, I, k lying on a dot. The radius of the 

 linear semicircle is to be nine intervals of the distance between the ruled 



,.2 



lines, so that each sqtuire on the paper has an area of ^-r. By counting the 



squares, the area glkng of the lower part acting towards the rib is found to be 

 the same as that of the upper area acting away from the rib ; each holds 



six squares, and gah = glkn = -^. And the height oi g = five intervals 



determines the limiting value of i„ to be such that cos\ = f, and i^ = 56'15. 



a sufficiently close approximation to 57"04°, the exact (Eankine's) value 



determined by the height of h where the curve crosses an. Both the linear 



semicircle AGB and abghi might be engraved on the squared paper ; for the 



r 

 superstructure — r above A the corresponding parallelogram is added, uo 



being drawn through u with the set-square, and |j gives a higher point of 



rupure. Now there is hoop-thrust at A, the crown, equal to that parallelo- 



wr"^ 

 gram --jr- Further, to allow for the extra density of the ring, the dotted 



treble boundary with hV =— p and kh' = j-r, and a like receding at / and g 



gives the increased thrust at the crown as measured by squares, or bj^ a 

 planimeter from fok'j)' , and the joint of rupture is 43'15°. If the arch-ring 

 widen ovX from A C, an allowance, always very slight, can be made by moving 

 to the right hotli the boundaries, as shown inside the arch. The line of stress 

 has to be moved into the middle of the arch-ring by a pair of equal and 

 opposite couples at the /o-im/s A andj/, and then modified by having a longer 

 radius of curvature at the A joint, due to the negative outward area f'aV 

 being left out, which causes the thrust at A to be greater by that amount. 



