PALMER: METHOD FOR CONSTRUCTING THE CATENARY. 225 
Substituting in the above equation we have 
4 * Cc OM OM 
oie (c = Ds ) 
hat 1s 
which is just the equation of our s-curve for the required catenary 
when passing through the given point. And c is simply the ratio 
of BM to PK or OM to OK, which ratio is easily obtainable by the 
use of the unit’s line as explained. Since FK is one unit DM is 
the ratio OM : OK. 
With the curve plotted, the level to which it drops is easily 
measured at once, and all other facts in regard to it readily derived. 
(6) Having given fixed abutments and the level to which the curve 
is to drop, to find the length of cord required and to plot the catenary. 
Here instead of the s-curve use the drawing of the ‘simple 
catenary.”’ Proceed just as in the first case, and find c, when the 
curve can be plotted at once and s found, and all the other features 
asin Pigrr. 
Fig. 8. 
Fig 8 shows fully the steps taken on the sheet of the simple ca- 
tenary. nese are precisely the same as in the case first con- 
sidered. 
Proof: For here we know that the required catenary, since it 
must go through the point of support B, will have an equation 
such that 
