PALMER: METHOD FOR CONSTRUCTING A CATENARY. 213 
Letting also 
w—weight of cord, per unit of length, 
W-=total weight of cord from O to any point P, and 
s—the length of the cord from O to P, 
we have the following relations regarding the piece of cord as a 
body in equilibrium: 
gs cosa. 
W=T sina ae) 
Wess. | 
From which, by dividing, 
Tsina | 2 dy W_ws (2) 
ide dar eT, 
BESTE) 
Differentiating, we have 
dy 
a(Z) 
yi Be a 
fey . |? ae) 
dx 
which integrates to 
fect (eyo } 6 
d 
But, since when x=o, =, we have C=o, and 
dy | Gy yee st ianwas 
or r-- (3°) =e Pons 
Transposing (=) and squaring, 
eis WX ( dy 
Io Tp —26 Tol) 
: j (3) 
From which 
d 2wx 
VV Oo —— Tis. 7 i = ae") 
(DHS -+ler Spee te eee (3) 
(eo) 
