PALMER: METHOD FOR CONSTRUCTING THE CATENARY. 223 
Discussion of the Cases. 
Preliminary to undertaking the treatment of these four cases, 
prepare as permanent diagrams which will serve for the solution 
of all problems that may be met so long as the same system of 
units is used, the following auxiliary drawings: 
Ci). On one sheet an accurate plotting of a catenary with the 
constant equal one, that 1s with equation 
This may be called the ‘‘simple catenary.” 
(2) On another sheet plot from the same construction work 
which gave the simple catenary a curve of values of s of this curve. 
That is obtain a plotting of 
I x ae 
s=— (c -e ) 
P 2 
by taking the difference of ordinates, Fig. 4, instead of the sum, as 
already explained. 
The one-half of each of these curves is sufficient. By means of 
these two plottings any problem in connection with the construc- 
tion of a catenary may now be treated. 
Constructions for the Four Cases. 
CASE I.—SUPPORTS OF THE SAME HEIGHT. 
(a) Having given a length of cord and fixed abutments to plot the 
curve of the cord. i 
