VOL. XIX. 3 I'HILOSOPHICAL TRANSACTIONS. IQl 



and CB. For because of the equiangled triangles J'fs and vfr, it is sf : rf :: 

 FR : VF. Or i : -^^+fL= ;: ^H^^y^^^T^ . ^^^ ^^ich therefore is equal to 



"_lil. Whence it is a : a -\- x :: a -\- x : vf, which also is the radius of a 



a 



circle equicurved with the catenaria in the point f. 



Carol. 4. — When the point f falls in a, or when the vertex is described by 

 evolution, that is, when x = O. the value of the evolving right line vf, which 



in this case is ka, becomes — "t- = a. That is, the point k, where the curve 



VK meets the axis, is as ituich above the vertex a of the chain, as c is depressed 

 below the same. Whence the diameter of a circle, equicurved vvich the chain 

 at its vertex, is equal to the axis of the conterminate hyperbola ah. Therefore 

 the cham ad and the h)perbola ah have the same degree of curvature at the 

 vertex a: for it is generally known that the aforesaid circle is equicurved with 

 the equilateral hy})erbola ah in ihe vertex a. Also this appears from the nature 

 of the chain itself, by what is demonstrated Prop. 1 of this. For the nascent 

 line FH, = ai' = the nascent bp = '^^ Sax, is double the nascent line bh or 

 y/lax i- XX = (when x vanishes) '^ 2ax. And therefore the same point is 

 both in the nascent hyperbola and the nascent catenaria. That is, the nascent 

 hyperbola ah coincides with the nascent catenaria ad, and therefore these lines 

 are equicurved at the vertex A. 



Curul. 5. — The curve kv is a third proportional to the right line ac, and the 

 curve af, or the right line al. For, from the nature of evolution, kv=vka 



— KA = vp — KA = — ■ a = o = . And there- 



fore a : Vl ax-\-x x :: '/2ax->rxx : K v. But ^^ la x-\- xx = \f, by Cor. 2, 

 Prop. 1. Whence ac : ap :: af : kv. 



Carol. 6.— The right line ki is double of ab. For since bi = bc=ca-I-ab, 

 it will be ai = ca + 2ab. But a k = c a, by Corol. 4 of this. Whence 

 KI = 2ab. 



Corol. 7. — The rectangle of ac and br is equal to double the hyperbolic space 



BAH. 



ForFR + AC = ^t-^-^- "'^ + " x« = « + ^X^2a:r + x.r = aX 



' ft 



Vlax-{-xx-\-a X ^ ■lax-\-xxz=: abXbh + acXbh = abXbh + acXbd 

 -f A c X D H. Therefore frXac — bdXac = bkXac=:abXbh + acX 

 D h. But by Prop. 4 of this, itisAcXDH = space a g f. And therefore b r 

 X ac = akhl + agf=2bah, by Cor. 1, Prop. 5. 



Prop. 7. Theor. — If in the logarithmic curve lag, fig. 6, pi. 4, whose given 

 sublangent hs is equal to the right line a,.(deteriiiined by Cor. 2, Prop. 2 of this) 



