Every catenary lying between them has its directrix higher, and every catenary lying beyond them has its directrix lower than that of the two catenaries.
The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary.
The only surface of revolution having this property is the catenoid formed by the revolution of a catenary about its directrix.
The condition of stability of a catenoid is therefore that the tangents at the extremities of its generatingcatenary must intersect before they reach the directrix.
Hence the tangents at A and B to the upper catenary must intersect above the directrix, and the tangents at A and B to the lowercatenary must intersect below the directrix.
This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.
To prove this, let us consider thecatenary as the form of equilibrium of a chain suspended between two fixed points A and B.
When the conic is a parabola the meridian line is a catenary (fig.
The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a cusp at the vertex of the catenary, and is asymptotic to the directrix.
Illustration] The surface formed by revolving the catenary about its directrix is named the alysseide.
One of the most laborious and practically useful works of Giddy was a treatise on the properties of the Catenary Curve.
You hesitate between the catenary and portion of a circle.
I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.
For a uniform catenary the limit was found above to be 1.
Finally, we may refer to the catenary of uniform strength, where the cross-section of the wire (or cable) is supposed to vary as the tension.
The catenary is not properly a curve capable of infinity, if its direction does not alter with its length; but it is capable of infinity, implying such alteration by the infinite removal of the points of suspension.
They have to be braced by cables, which make catenary curves like suspension-bridges on end.
There were tall buildings, but their walls were draping, catenary curves.
Soon after recrossing the snow catenary already mentioned we quitted the ridge to get obliquely along the slope of the pyramid.
How to pass this snow catenary I knew not, for I did not think a human foot could trust itself upon so frail a support.
To this catenary the trolley wire was suspended by hangers at short intervals.
There must be two attached alternately to thecatenary cable.
This truly remarkable showing can only be explained by the catenary action of the main steel, and some truss action by the steel which was horizontal, in conjunction with the U-bars, of which there were plenty.
If steel be placed in this manner, the catenary curve will be practically approximated, the steel will be fairly developed throughout its length of embedment, and the structure will be proof against cracking.
In his advocacy of placing steel to simulate a catenary curve, with end anchorage, the author is more nearly correct than in other issues he makes.
