PALMER: METHOD FOR CONSTRUCTING THE CATENARY. 217 
II. A parabola, with vertex at O”, Fig. 1, and focus at O, when 
rolled on O”X will generate a catenary, the focus O tracing the 
curve. 
Proof: 
Let OA, Fig. 3, be. a parabola with focus at F.. -If the focus of 
the parabola traces the catenary FP as the parabola rolls on its 
tangent ON, at O, it must be that when F is at any point P on the 
catenary, arc OA of the parabola must equal ON, FA must equal 
the radius of curvature of the catenary at P, PN; and FR, the per- 
pendicular from F upon the tangent to the parabola at A must 
equal PM, or y of the catenary. 
Call the codrdinates of A on the parabola (x’,y’) and those of the 
point P on the catenary (x,y). Call the radius of curvature of the 
catenary p, and the angle the tangent makes at A, a. 
Now, from the properties of the parabola, RA bisects Z FAD, 
Ae Poe Ris always on-ON; and Z REA= ~” KAT =<: 
Then from the figure p—y +c (16) 
tanta Manes Be Eas 
x y 
and 
ee for the parabola x? acy is pie) 
dx’ \ c 
p see ee (17) 
| ¥* 
which gives 
