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VII. On the Catenary Ctirve. By A Correspondent. 



To Mr. Tilloch. 



Sir, — X HE following problem relating to the catenarian curve 

 will not be uninteresting to some of your readers, if you can find 

 room for it. 



Prob. Of all the catenarian curves that can be formed by 

 suspending different lengths of the same chain of uniform thick- 

 ness, from two given points placed in the same horizontal line, 

 it is required to determine that one in which the pull at the 

 points of suspension is least. 



Let X represent the absciss taken from the vertex, or lowest 

 point of the curve, along the vertical axis ; y the corresponding 

 horizontal ordinate ; and z the length of chain, or the part of 

 the curve line between the ordinate y and the vertex. Then 

 the weight of the chain x is sustained by the two pulls at its ex- 

 tremities. The pull at the vertex, denoted by a, is horizontal 

 in its direction. The pull at the other extremity of z, denoted 

 by f, is oblique to the horizon j and, by the resolution of forces, 



it is equivalent to the force — — ^ x y, acting horizon- 



^rlx' + dy- 



tally in contrary direction to a; and likewise to the force, 

 — — X f^ directed vertically upwards. Now the equili- 



*/ (bfi + dy- 



brium of every part of the chain requires that the horizontal and 

 vertical forces acting upon it in opposite directions, shall be se- 

 parately equal to one another : wherefore, 



-=^=:x/=a (1) 



/^dx'^ + dy'' 



dx /. 

 X /= 2. 



,^/dx- + di/- 



From these equations, we get 



dx : dy : '. z : a ; and hence 

 -v/ dx^ 4- dy"- = dz : dy : : V z^ -\- a"- : a 

 J^z=i d: 

 dy = 



»/ dx^ -\- dy^ z=. dz : dx '. : -v/x* + o' : z ; wherefore 



adz 



dx = 



■.dz ' 



and if we now integrate, observing that x, ?/, and z vanish to- 

 gether, we shall obtain 



'>! = 



