320 THE STONE AECII. [CHAP. XV. 



from the nearest side on a line parallel to the other two, gives 

 the position of the fourth side of the parallelogram for which 

 P is upon the kernel. 



173. The resultant prcsurc should therefore act 

 within the middle third of the joint area. As this prin- 

 ciple is most important, and the demonstrations of Chapter VI., 

 upon which the above result is based, may appear to some too 

 purely mathematical, we give here the demonstration of the 

 same principle as given by Wbodbury, in the work above 

 cited. 



" Suppose the pressure to be nothing at the intrados #, and 

 to increase uniformly from that point to the extrados b (PL 

 24, Fig. 96). It is plain that the pressure at any point along ab 

 will be represented by the ordinate of a certain triangle. The 

 whole pressure will be represented by the surface of that tri- 

 angle ; and the point of application of the resultant of all the 

 pressures will be at c opposite the centre of gravity of that 

 triangle. We have then c b = % a b. Vice versa, if the point 

 of application be at c, G b = $ a b, we know that the pressure 

 is nothing at a. 



" If the point of application be at c, G b being less than a 5, 

 G being still opposite the centre of gravity of the triangle 

 whose ordinates represent the pressure, we know that the ver- 

 tex of that triangle and point of no pressure are at e, b e = 3 

 xbc. 



" In this case, the joint a b will open at a as far as e ; the 

 adjacent joints will also open until we come to one where the 

 curve of pressure passes within the prescribed limit. 



" This reasoning is, of course, applicable to all the joints ; 

 and we readily conclude that the curves of pressure should lie 

 entirely between two other curves which divide the joint into 

 three equal parts." 



Thus, in PI. 24-, Fig. 97, suppose the resultant P of l the 

 upper part of the wall to have the position as represented, so 

 that it intersects the joint B D in C outside of the middle third 

 of the cross-section. The entire pressure is distributed over 

 3 C B = A B, and the area D A does not act at all. Moreover, 

 the pressure at B is twice as great as when P passes through 

 the centre of gravity and is uniformly distributed over A B, or 

 is fds of the uniformly distributed pressure of P upon C B. 



Beyond A the pressure is zero, and the conditions of load 



