Alexander an'd Jackson — Pohigons to Generate Diagrams, Sec '2t 



As the squave-dressed heavy backing is built up from ol', its power of 

 resisting the spreading of the arch from point to point as it rises is deter- 

 mined by the shape of pMco. This is a good approximation, taking the shape 

 of the spandril-area parabolic, and the excess load along the rib as if it were 

 along the span. 



Exact Solution of the Mc(sonry Semicircv.lar Arch. 



The quadrant ABS has three loads a parallelogram of height AD, the 

 spaudril area and a load spread uniformly along the arc ABS on fig. 4«, 

 where it is shown like a collar 3IIIA, of a depth one-half of the radius. 

 Such might be the actual case in a brick sewer, whose ring was deep, and 

 heavier Ihau the surrounding mass. We have chosen this extreme depth h' 

 so that the area of the collar bit by bit of the arc shall correspond to the 



Surface L o f C Fl uid 



Fig. 4a. 



sectors subtended at by those arcs. So now the whole load on the arc AB is 

 the trapezoid OBGD. It at once becomes evident that the horizontal conjugate 

 load-area is an isosceles right-angled triangle LUG, mapped out by the 

 vertical LH, and a 45° or 1-to-l slope LEG. The horizontal load on the 

 arc SB is the trapezoid GFEH. aSTow, the two trapezoids P and Q have their 

 parallel sides ec^ual each to each, and so P : Q = CD : EH = sin : cos Q, and 

 T is tangential to the rib, and the rib is balanced.* Although there is always 



* It will be seen that if the shell ABSaM have the same weight as the water (or fluid 

 as mercury) it will be balanced by the external fluid at auy depth, the flu.d supplying 

 the two conjugate loads simultaneously, and sinking, merely adding equal parallelograms 

 to each. 



This elegant theorem of the equiliVirium of the thin horizontal empty circular 

 cylinder displacing its own weight of fluid was given by the authors in a letter to 

 " Nature " of 18th February, 1897. From some private correspondence about it with 



