188 PHILOSOPHICAL TRANSACTIONS, [aJJNO \6gT- 



For by the construction it is b k = —=f^=. Therefore the fluxion of 



v2a.r + .rx 



the space abkrla is BK/ib = bk x b/j = ""■*' — ^ a„(J since bf = 



■v2ax + xx 



:; — , and Ac is given; its nuxion will be bf = ^- — =. But in the 



construction of the foregoing proposition, the fluxion of the ordinate bf = 



— . Wherefore this construction comes to the same as the construc- 



V2ax + xx 



tion of the foregoing proposition, and consequently the point f is at a catena- 



ria. Q. E. D. 



Corol. — As in the foregoing proposition, the catenaria is described from 



the given length of the parabolic curve ; so in this its description depends 



on the quadrature of the space, in which a:xyy=::a^ — laxyy. ForBKorj/^ 



Vawx + XX 



Prop. IV. Theor. — The space agf (fig. 4) contained by the catenaria af, 

 and the right lines fg, ag, parallel to ab, bf, is equal to the rectangle under 

 the semiaxis ac, and dh the distance of the ordinates in the hyperbola and 

 catenaria. 



For DH = BH — BD = (by Prop. 2 of this) "^ ^^ , "^ ■ = 



'^ J ^ ' V2ax + xx V2aj+.-. 



—. ■ Wherefore the fluxion of the rectangle under the given line ac 



and HD is , = ,r x -7== =: s f X f g = the fluxion of the space 



V2ax + xx V'2ax-i-xx "^ "^ 



agf. And since these figures are nascent together, it follows that the rect- 

 angle under ac and dh is equal to the space agf. q. e. d. 



Corol. Hence it follows that the space fad, comprehended by the chain 

 fad and the horizontal right line fd, is equal to the rectangle under fd and 

 BA, subtracting the rectangle under either axis of the hyperbola ah, and dh 

 the excess of the right line bh, or of the curve ad, above the ordinate bd. 



Prop. V. Theor. — If the rectangle le, equal to the hyperbolic space alh, 

 be applied to the right line al, the point e will be the centre of equilibrium of 

 the catenarian curve afd. 



Let a heavy curve fa be conceived to be poised on the axis gl. From the 

 doctrine of the centre of gravity it follows, that the moment of the weight 

 fa is expounded by the surface of an upright cylinder erected on fa, and cut 

 ofi^ by a plane passing through gl, making half a right angle with the plane 

 of the curve. And the fluxion of this surface, or faXfg, is equal to the 

 fluxion of the space alh, or bhXhl: because fa and bh, as also fg and 



