The arc of the centenary y=ccosh(x/c) from the vertex to the point (x,y).

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Let,

(x,y) be any point on  catenary. The length of the arc to the point is  s and linear density is wo

Hence the weight of the gravity can be wos

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.......... ( A )

wos = T sin θ........ ( B)

Dividing ( B) by ( A ) 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.