Hello.

The catenary is a plane curve, whose shape corresponds to a hanging homogeneous flexible chain supported at its ends and sagging under the force of gravity.

Let,

( x,y ) be the arbitrary point on the catenary.

s be the length of arc from the lowest point of catenary to the point ( x, y )

wo be the linear density of the catenary.

So, weight of gravity = wos

To be the tension at lowest point

T1 be the tension at ( x, y )

Hence,

Resolving the tension forces in X and Y directions, we get, the tension in X directions,

To = T cosθ.......... ( 1 )

wos = T sin θ........ ( 2 )

Dividing ( 2 ) by ( 1 ) we get,

tan θ = wos / To

So, s = To . tan θ / wo

Let, To / wo = c

Hence, s = c. tanθ........ ( 3 )

The above equation ( 3 ) is known as the intrinsic equation of the Catenary.

Best Wishes.