A ﬂexible inextensible string is suspended from two ﬁxed points, and takes up a position of equilibrium under the action of gravity. It is required to ﬁnd the equation of the curve in which it hangs.
Let be the weight of unit length, and the length of arc measured from the lowest point ; then is the weight of the portion . This is balanced by the terminal tensions, acting in the tangent line at , and in the horizontal tangent. Resolving horizontally and vertically gives
in which is the inclination of the tangent at ; hence
where is written for , the length whose weight is the constant horizontal tension; therefore
which is the required equation of the catenary, referred to an axis of drawn at a distance below .
The following trigonometric method illustrates the use of the gudermanian: The “intrinsic equation,” , gives ; hence ; ; thus ; whence ; and .
Numerical Exercise.—A chain whose length is 30 feet is suspended from two points 20 feet apart in the same horizontal; ﬁnd the parameter , and the depth of the lowest point.
The equation gives , which, by putting , may be written . By examining the intersection of the graphs of , , it appears that the root of this equation is , nearly. To ﬁnd a closer approximation to the root, write the equation in the form , then, by the tables,
whence, by interpolation, it is found that , and , . The ordinate of either of the ﬁxed points is given by the equation
from tables; hence , and required depth of the vertex feet.14