DR WALLACE ON A FUNCTIONAL EQUATION. 



637 



thickness at the crown in respect to the rise of the arch. From the catenary, 

 however, we can construct an equilibrated arch that shall have any height and 

 span and thickness at the crown that may be requu-ed. Hence a table of co-ordi- 

 nates of the catenary will serve for the construction of bridges which are rigid in 

 all their parts, as well as for bridges of suspension. In fact, curves of suspension 

 and the catenary belong to the same family of curves, and are nearly related. 

 Tlie nature of the former is expressed by a formula involving two parameters ; 

 but, when these are supposed equal, the general analytical expression for a curve 

 of equilibration becomes the equation of the catenary. 



24. To avoid reference to any but the most elementary theories, I shall be- 

 gin with investigating a property of an equilibrated polygon. 



Equilibrated Polygon. 



Problem. 



Fig. 3. 



Let ABCDEFGH be a chain formed by straight rods of any length, which 

 turn with perfect freedom about their extremities as joints. Suppose that the 

 chain hangs vertically from two fixed points. A, H ; and, abstracting from its own 

 weight, that it is loaded at the joints with given weights, or masses of matter. 

 It is proposed to determine the geometrical condition that must be satisfied by 

 the position of the rods when the whole constitute an equilibrium. ' 



Let BC, CD, DE be any three adjoining rods ; these would manifestl}'^ form 

 an equilibrium, independently of the others, if the extreme points B and E were 

 fixed, the links BC, ED turning freely about them as centres : Produce BC, and 

 ED, the extreme links of these three, until they meet in K : Let m and m' de- 

 note the weights on the chain at C and D, and let be their centre of gravity. 



The rod CD is urged by three forces, viz. two in the directions KB, KE, and 

 the vertical pressures of the masses m and m', which are equivalent to a single 

 mass pressing the rod vertically downward at their centre of gravity. The di- 

 rection of this last force must, in the case of equilibrium, pass through K, the in- 



VOL. XIV. PART II. 6 c 



