216 KANSAS UNIVERSITY QUARTERLY. 
and, differentiating, bearing in mind equations (7) and (6) 
GENO Y. 
Gane sc (13) 
Substituting in the above expression 
Cc? is y : ss PUY 
Sey a) 
c2 
pp 
This is represented by the length of the normal at point P. be- 
tween the point and the directrix, that is by PN, Fig. 1. 
For PN, cossa=-y— PN : by the figure and (10). 
y? 
PN=-——=p. 
ae Tei 
Hence the radius of curvature for any point is readily obtained. 
Area Onder the Curve. 
A Syax= J Se ; Beye o )ax 
A= =a ; (15) 
That is, in: Fig. r, the area O”OPM=2> the triangle MFP, or 
—a rectangle with c and s for sides. 
And the area OPE—cs—cx—c(s—x). 
Other Properties. 
Other properties that may readily be demonstrated directly from 
the foregoing are: 
I. The Involute of the Catenary is the tractrix. 
Point F, Fig. 1, is always on the involute of the catenary from 
the vertex. For FP is constantly equal s, the length of curve OP. 
FM is constantly tangent to this involute and is of constant length 
equal c, which is O”O of the figure. The curve, which is the locus 
of F, is hence the equi-tangential curve, the tractrix. 
