PALMER: METHOD FOR CONSTRUCTING THE CAYENARY. 227 
D ee 
] P=v=e(e 2c —e x), (21) 
in which 1, v and D are given quantities, and c is the parameter of 
the desired catenary. Inspecting this equation we see that it is of 
just the form of the s-curve, so that c may be obtained from this 
equation by the same steps as taken in Case I (a), merely noticing 
that the result derived by these steps will here be twice c, and 
hence must be divided for the value of c to use in plotting the de- 
sired catenary. Simply construct a right triangle to scale, with | 
for hypotenuse and v for one side. The other side will at once be 
V 1?v?, and should be used precisely as s, was in Case I (a). 
Then D is to be used as x, was, when the resulting value will 
be 2c! 
Proof of Equation: Referring to Fig. g 
Kine 
Bet x4 —-xk>—K: (a constant) 
D+k 
x,=—— 
2 
D—k 
X,=——__, 
2 
C isd us Teh X2 ARE 
Missle ar aa (c eye POSSE eae c ) 
Now substitute above values of x, and x, and obtain 
Civ panier Die all aetiheceal Shin DS ee 
== ( 2 eze—e 2e 2x+terxe %2#—e 2c @ 2e ) 
2 
iG D k D US YOY nila 2 AK ities D k 
v=—(é 2c @ 2c +6 zc & ze —@ 2c @ zc —@ ne © 2c ) 
2 
From which 
