218 KANSAS UNIVERSITY QUARTERLY. 
Equation (14) for the catenary gives the radius of curvature p 
equal -—. Therefore the curve traced by the focus of the parabola 
c 
is the catenary. 
III. The center of gravity of the catenary is lower than that of 
any other curve which can be formed from the same length of line 
between the two fixed points in space. If this were not so the 
cord would of its own weight fall in to the form having a lower 
center of gravity. 
The center of gravity for any catenary may be located at once, 
very easily, thus; Fig. 1. 
Bisect ON. From this middle point of ON draw a parallel to 
PN. When this line cuts OY is the center of gravity, 
For by the usual method of mechanics the height of the center | 
of gravity of the catenary above the X axis 1s 
= I Cx, 
y=—|(yit =}, 
2 S, 
and by similar triangles it can easily be shown that NP produced, 
Fig. 1, cuts OY produced in a point which is at a vertical distance 
5 CX 
above y,, or point P, equal to 
Pad 
PLOTTING THE CURVE. 
CATENARY THE SUM OF TWO EXPONENTIAL CURVES. 
Analyzing the equation of the catenary 
it will be noticed that the ordinate y may be considered as the sum 
of two portions y and y”, such that 
f Cc 3s 
bigs hn 
io soe x 
Vy =e Gix (18) 
: CG th a f - 
Now y’—=—-e « is recognizable at once’ as the well known ex- 
2 D 
ponential or logarithmic curve. And y” is plainly the ordinate of 
the same curve, at the same x distance from the vertical axis on 
the opposite side of the origin, 
