EQUILIBRIUM OF THE ARCH. 311 



If therefore the settlement be considerable, we may conclude that 

 the line of pressure touches the extrados at the crown, and for 

 some distance on either side of it. The material of the arch may 

 therefore be expected to yield more particularly about that point and 

 the points Q and Q' than any other; a great proportion of the 

 pressure being there thrown upon the edges of the voussoirs. 



11. If by reason of such yielding, or from any other alteration in 

 the forces impressed upon the mass, or in the circumstances of their ap- 

 plication, the form of the line of pressure be altered, it may manifestly 

 be expected to intersect the surface of the mass first about those points; 

 the least possible alteration of form being there sufficient to produce 

 the intersection. And this being the case, the portion of the arch above 

 Q and Q' must separate into two portions, revolving at those points 

 about the lower portions of the arch (see fig. 5) and at A, upon the 

 extremities of one another. 



Nevertheless this revolution is manifestly impossible unless the 

 points Q and Q yield outwards. And this can only take place by 

 the yielding of the material at Q and Q', by the slipping back of 

 the voussoirs there, or by the portions of the arch or its abutments 

 beneath those points revolving outwards, in consequence of the inter- 

 section of the extrados by the extremities QR and QR' of the line 

 of pressure (fig. 3). 



The last is in point of fact the cause which leads, in the great 

 majority of cases, to the fall of the arch. 



The extremity R of the line of pressure is made to cut the 

 extrados of the arch, or the outer surface of the pier, by the 

 diminution or removal of some force which acted there in opposition 

 to the tendency of the arch to spread itself, and which kept the 

 direction of the line of pressure within its mass, — the resistance of 

 a mass of earth for instance, or the opposite thrust of some other 

 arch springing from the same pier or abutment. 



On the whole, then, it appears that in the commencement of its 

 fall the arch will divide itself into six distinct portions, of which four 



