VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 189 



HL, are equal. And therefore, since they are nascent at the same time, the said 

 superficies of the erect cylinder is equal to the hyperbolic space ahl. There- 

 fore this applied to the heavy line itself af, or to the right line al, wliich is 

 equal to it, it produces a breadth ae equal to the distance of the centre of gra- 

 vity from the axis of libration gl. Hence e will be the centre of equilibrium 

 of the curve fad, lying equally on each side of the axis ab. q. e. d. 



Corol. 1 . — The spaces abhl, bah, and agf, are in arith metical proportion, 



a.v- + XX a.i+ r.i x i' 



For the fluxion of the space alh = ,- X x = -7=-— — = 



2ax+ xx_ ax x x _ ^, ^ 2 « J. + ,r X -J'^f^= = fluxion of the space bah, les- 



V'iax + xx V2<3x + jx 



sened by the fluxion of the space agf, by prop. 4 of this. And as these three 

 figures are nascent at the same time, it will be bah — agf = (alh =)el — 

 BAH. So that 2 BAH = BL + AGF. Whencc it follows that the spaces bl, bah, 

 and AGF, are in arithmetical proportion. 



Corol. 1. — The centre of gravity of the catenaria descends lower than that of 

 any other line of the same length, and having the same extremities. For every 

 heavy body descends as low as it can. And since a figure descends just so much 

 as its centre of gravity descends, a heavy flexible line will so dispose itself, as 

 that its centre of gravity will be lower than if it assumes any other figure. And. 

 from this property of a heavy flexible line all its other properties might be easily 

 deduced. 



Corol. 3. — If upon any curves, having the same length and the same limits 

 D and F as the catenaria pad, upright cylinders were cut by a plane passing 

 through DF ; of the cylindrical superficies so cut off", the greatest is that which 

 insists on the catenaria. For these superficies, if the angle made by the planes 

 be half a right angle, applied to the curves themselves, which in the present 

 case are of the same length, produce breadths equal to the distances of the 

 centre of gravity of the curves from the right line df. Now as, in the CMte- 

 naria this distance is the greatest, because of the greatest descent of the centre 

 of gravity, the cylindric surface to be applied, will also be the greatest. And 

 because there is the same ratio of c)lindrical surfaces cut oti' by a plane, con- 

 taining any angle with the plane of the base, as when the said angle is half a 

 right angle, the proposition obtains universally. 



Ltmnia. — If upon any ordinate fb (fig. 4) perpendicular to the axis ab of any 

 curve APa, that is described by the evolution of ancAlier curve kv, from the 

 corresponding point v in kv a perpendicular vr be let fall, meeting the ordinate 

 in R ; if the fluxion of the axis ab remains the same, the fluxioa of the fluxion 



