thumb|Diagram of forces acting on a segment of a catenary from to . The forces are the tension at , the tension at , and the weight of the chain . Since the chain is at rest the sum of these forces must be zero. A differential equation for the curve may be derived as follows.Following Routh Art. 443 p. 316 Let be the lowest point on the chain, called the vertex of the catenary.Routh Art. 443 p. 317 The slope of the curve is zero at since it is a minimum point. Assume is to the right of since the other case is implied by symmetry. The forces acting on the section of the chain from to are the tension of the chain at , the tension of the chain at , and the weight of the chain. The tension at is tangent to the curve at and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written where is the magnitude of the force. The tension at is parallel to the curve at and pulls the section to the right. The tension at can be split into two components so it may be written , where is the magnitude of the force and is the angle between the curve at and the -axis (see tangential angle). Finally, the weight of the chain is represented by where is the weight per unit length and is the length of the segment of chain between and .
thumb|Diagram of forces acting on a segment of a catenary from to . The forces are the tension at  the tension at , and the weight of the chain . Since the chain is at rest the sum of these forces must be zero. A differential equation for the cu,rve may be derived as follows.Following Routh Art. 443 p. 316 Let be the lowest point on the chain, called the vertex of the catenary.Routh Art. 443 p. 317 The slope of the curve is zero at since it is a minimum point. Assume is. to the right of since the other case is implied by symmetry. The forces acting on the section of the chain from to are the tension of the chain at , the tension of the chain at , and the weight of the chain. The tension at is tangent to the curve at and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written where is the magnitude of the force. The tension at is parallel to the curve at and pulls the section to the right The tension at can be split into two components so it may be written , where is the magnitude of the force and is the angle between the curve at and the -axis (see tangential angle). Finally, the weight of the chain is represented by where is the weight per unit length and is the length of the segment of chain between and .