VOL. XIX. 3 I'HILtSSOPHlCAL TRANSACTIONS. 187 



part of the same acting according to the direction dJ', as the semiaxis of the 

 conterminate hyperbola ah, is to da the length of the chain to the vertex of 

 the curve. Whence, when the chain is given, this ratio is given. And in the 

 same chain suspended more or less loosely, that horizontal force is as the axis 

 of the conterminate hyperbola, since da remains the same, when the extremes 

 of the chain are equally high. 



Carol. 6. — In a vertical plane, but in an inverted situation, the chnin will 

 preserve its figure without falling, and therefore will constitute a very thin 

 arch or fornix : that is, infinitely small, rigid, and polished spheres, disposed 

 in an inverted curve of a catenaria, will form an arch, no part of which will be 

 thrust outwards or inwards by other parts, but, the lowest parts remaining 

 firm, it will support itself by means of its figure. For since the situation of 

 the points of the catenaria is the same, and the inclination of the parts to the 

 horizon, whether in the situation fad, or in an inverted situation, so that the 

 curve may be in a plane which is perpendicular to the horizon ; it is plain, that 

 it must keep its figure unchanged as well in one situation as the other. And 

 on the contrary, none but the catenaria is the figure of a true and legitimate 

 arch or fornix. And when arches of other figures are supported, it is because 

 in their thickness some catenaria isinckided. Neither would it be sustained, if 

 it were very thin, and composed of slippery parts. From Corol. 5, before, it 

 may be collected, by what force an arch or buttress presses a wall outwardly, 

 to which it is applied. For this is the same with that part of the force sus- 

 taining the chain, which draws according to a horizontal direction. For the 

 force which in the chain draws inwards, in an arch equal to the chain drives 

 outwards. All other circumstances, concerning the strength of walls to which 

 arches are applied, may be geometrically determijied from this theory, which 

 are the chief things in the construction of edifices. 



Corol. 7- — Instead of gravity, if any other power exerts its force, acting in 

 like manner on a flexible line, the same curve will be produced. For example, 

 if the wind be supposed equable, and should blow according to right lines 

 parallel to a given line ; the line thus inflated by the wind would be the same 

 as the catenaria. For since all things obtain in this other force, as we have 

 supposed in gravity, it is evident, the same line must be produced. 



Prop. III. Theor. — The hyperbola aforesaid ah remaining, (fig. 5, pi. 4,) 

 if through a, a right line gal be drawn perpendicular to the axis ab, and a 

 curve KR be described of such a nature, that bk may be a third proportional to 

 the right lines ba and AC, and to ac be applied a rectangle av equal to the in- 

 terminate space abkkla ; the concourse f of the right lines hb, vg, will be at 

 a catenaria. 



B B 2 



