124 LECTURE XIV. 



by the weight of the portion of the chain below the link, the other by the 

 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 consti- 

 tute 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 equilibrium renders an accurate 

 experimental proof of the proposition somewhat difficult ; but it may be 

 shown that a slight degree of friction is sufficient for retaining in equili- 

 brium 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 equal ; the same 

 method may be applied to the determination of the form requisite for an 

 equilibrium, whatever 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 circumstance into the experiment, by suspending 

 proportional weights from different parts of the chain. Thus we may 

 employ wires 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 form 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 XI. 

 Fig. 152... 154.) 



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

 vertical direction, is by no means perfectly applicable to cases which 

 generally occur in practice. The pressure of loose stones and earth, 

 moistened as they frequently are by rain, is exerted very nearly in the same 

 manner as the pressure of fluids, which act equally in all directions : and 

 even if they were united into a mass, they would constitute a kind of 

 wedge, and would thus produce a pressure of a similar nature, notwith- 

 standing the precaution recommended by some authors, of making the 

 surfaces of the arch stones vertical and horizontal only. This precaution 

 is, however, in all respects unnecessary, because the effect which it is 

 intended to obviate, is productive of no inconvenience, except that of 



* For its properties see D. Gregory, Ph. Tr. xix. 637, and xxi. 419. Clairaut 

 on Catenariae, Miscellanea, Berolin, 1743, vii. 270. Krafft, Novi Com. Petrop: v. 

 145. Cantezzani, Com. Bon. vi. O. 265. Legendre, Mem. de Paris, 1786, p. 20. 

 Fuss. N. A. Pet. 1794, xii. 145. The elementary works of Poisson, Traite de Me- 

 canique, and Whewell's and Earnshaw's Mechanics. 



