214 KANSAS UNIVERSITY QUARTERLY. 
which is the general equation of the Catenary, referred to O as the 
origin. 
By shifting the origin to O”, a distance a below O, the con- 
stant of integration disappears, and we have the form 
awa : 
j= . "(e To +e T.*), (5) 
w mi 
: : 
For convenience, represent —* in the equation by c. Then 
Ww 
y= ; ‘Gc Ti a ), (6) 
which is the usual form for the equation. 
ANALYTICAL PROPERTIES OF THE CURVE. 
Combining equations (2) and (3) we have 
or 
x x 
s—“(e iO ae e ). (7) 
Squaring (6) and (7) and subtracting, 
y?=s?-c?; (8) 
and equation (2) is 
Ss 
tan oat (9) 
According to (8) and (9g), then, we may draw a tangent at any 
point on the curve, or find at once, graphically, the angle a, Fig. 1, 
and also the length of the curve from O to the point, when we have 
the parameter c known. it is only necessary to draw a semi-circle 
upon y at the point, and strike an arc with c asa radius from M, 
Fig. 1, thus forming a right triangle with y, s and c for sides, and 
determining the tangent at P. Then from the figure, if 
