EQUILIBRIUM OF THE ARCH. SOI 



First. That certain of the sections, which we have imagined, be- 

 come real sections of the mass, dividing it into separate and distinct 

 parts, each of which retains the properties of a perfect solid. 



Secondly. Let us suppose every point in the system to admit of 

 displacement, subject, within certain limits, to the law of perfect 

 elasticity. 



The determination of the conditions of the equilibrium in these 

 two cases, will constitute a complete theory of construction. 



The discussion contained in the remainder of this paper will be 

 confined to the first case. 



6. Let the mass AB (fig. 2.) have for its line of pressure the 

 line PP'. Now it is clear, that if this line cut the plane QQ of any 

 section of the mass in a point n' without the surface of the mass ; 

 the tendency of the opposite resultants of the forces acting upon the 

 two parts AQQ and SQQ', into which that section divides the mass, 

 will be to cause them to revolve about the nearest point Q' of its 

 intersection with the surface of the mass. And, this tendency being 

 wholly unopposed, motion will ensue. And so in the mass represented 

 (fig. 6.) the force p and with it the line of pressure pp' being given, 

 it appears that, being cut transversely as shewn in the figure, the mass 

 cannot be supported by any single force p if it extend beyond CD': 

 any such force must, to produce equilibrium, be applied at q; and 

 being applied there, the portion C'C"Z)"'iy will be wholly unsupported. 

 The line of pressure being continued cuts the planes of the sections 

 CD', CD', &c., without the surface of the mass. 



Thus then it is a condition of the equilibrium, that the line of 

 pressure should intersect the plane of every section of the body within 

 its mass. 



This condition will be satisfied if this line nowhere cut the surface 

 of the mass except at the points P and P. Fig. 2. Or if the equation 



■^F^z, F^z, ■ « = 0, 



RR Z 



