ON ARCHITECTURE AND CARPENTRY. l6l 



each link is held in equilibrium by the result of the two forces acting at its ex- 

 tremities ; and these forces or tensions are produced, the one by the weight of 

 the portion of the chain below the link, the other by ^:he same weight increased 

 by that of the link ; both of them acting originally in a vertical direction. Now 

 supposing the chain inverted, so as to constitute an arch of the same form 

 and weight, the relative situations of all the lines, indicating the directions of 

 the forces, will remain the same, the forces acting only in contrary directions, 

 so that they are compounded in a similar manner, and balance each other on 

 the same conditions, but with this difference, that the equilibrium of the 

 chain is stable, and that of the arch tottering. This property of the equili- 

 brium renders an ;iccurate experimental proof of the proposition somewhat 

 difficult ; but it may be shown that a slight degree of friction is sufficient for 

 retaining in equilibrium an arch formed by the inversion of a chain of beads. 

 The figure is called a catenaria,when the links are supposed to be infinitely small, 

 and the curvature is greatest at the middle of the chain. It is not at all necessary 

 to the experiment that the links of the chain be ecjual; the same method may 

 be applied to the determination of the form requisite for an equilibrium, what- 

 ever may be the length or weight of the constituent parts of the arch ; and when 

 the arch is to be loaded unequally in different parts, we may introduce this cir- 

 cumstance into the experiment, by suspending proportional weights from 

 different parts of the chain. Thus we may employ Avires or other chains to 

 represent the pressure, and adjusting them by degrees, till their extremities 

 hang in a given line, we may find the form which will best support the weight 

 of the materials, the upper surface,or extrados,of the arch being represented by 

 the same line in an inverted position, while the original chain shows the forni 

 of the intrados, or of the. curve required for the arch stones themselves. In 

 common cases, the form thus determined will differ little from a circular arc, 

 of the extent of about one third of a whole circle, rising from the abutments 

 with an inclination of 30° to the vertical line, and it never acquires a direction 

 much more nearly perpendicular to the horizon. It usually becomes more 

 curved at some distance below the summit, and then again less curved. (Plate 

 XL Fig. 152 . . 154.) 



But the supposition of an arch resisting a weight, which acts only in a ver- 

 tical direction, is by no means perfectly applicable to cases which generally 

 occur in practice. The pressure of loose stones and earth, moistened as they 



VOL. I. y 



