# Definition:Abel's Integral Equation

## Definition

**Abel's integral equation** is an integral equation whose purpose is to solve Abel's mechanical problem, which finds how long it will take a bead to slide down a wire.

The purpose of **Abel's integral equation** is to find the shape of the curve into which the wire is bent in order to yield that result:

Let $T \left({y}\right)$ be a function which specifies the total time of descent for a given starting height.

- $\displaystyle T \left({y_0}\right) = \int_{y \mathop = y_0}^{y \mathop = 0} \, \mathrm d t = \frac 1 {\sqrt {2 g} } \int_0^{y_0} \frac 1 {\sqrt {y_0 - y} } \frac {\mathrm d s} {\mathrm d y} \, \mathrm d y$

where:

- $y$ is the height of the bead at time $t$
- $y_0$ is the height from which the bead is released
- $g$ is Acceleration Due to Gravity
- $s \left({y}\right)$ is the distance along the curve as a function of height.

## Also see

## Source of Name

This entry was named for Niels Henrik Abel.

## Historical Note

Niels Henrik Abel devised what is now known as Abel's Integral Equation as a tool by which to solve the Tautochrone Problem in $1823$.