EQUILIBRIUM OF THE ARCH. 809 



It is manifest that the converse of this operation is possible. 



9. Having given the form and position of the line of pressure, and 

 the positions of the different sections to be made through the mass, we 

 may, for instance, enquire what form these conditions impose upon the 

 surface which bounds it. 



Or we may make the direction of the line of pressure and the 

 form of the bounding surface subject to certain conditions not abso- 

 lutely determining either. '"* t'^ oxi'>»^i'« inn ouJ ifion 



For instance, if we suppose the form of the intrados of an arch to 

 be given, and the direction of the intersecting plane to be always per- 

 pendicular to it, and if we suppose the line of pressure to intersect this 

 plane always at the same given angle with the perpendicular to it, 

 so that the tendency of the pressure to thrust each from its place may 

 be the same, — we may determine what under these circumstances must 

 be the extrados of the arch. ,''^'^",' '"'^ 



If this angle equal constantly the limiting angle of resistance, the 

 arch is in a state bordering upon motion, each voussoir being upon 

 the point of slipping downwards or upwards, according as the constant 

 angle is measured above or below the perpendicular to the surface of 

 the voussoir. 



The systems of voussoirs which satisfy these two conditions are the 

 greatest and least possible. 



If the constant angle be zero, the line of pressure being every- 

 where perpendicular to the joints of the voussoirs, the arch would 

 stand even if there were no friction of their surfaces. 



It is then technically said to be equilibrated. It is impossible to 

 conceive any arrangement of the parts of an arch by which its stabi- 

 lity can be more effectually secured*. 



10. The theory stated above readily explains the phenomena ob- 

 served in the settlement and fall of the arch. 



* The great arches of late years erected by Mr Rennie, in this country, have for the 

 most part been so loaded as very nearly to satisfy this condition. 



ss 2 



