384 
PKOFESSOR J. CASEY ON A NEW 
and in subsequent sections ; for this reason, and also on account of its extremely 
elementary character, we give here an investigation of the leading properties of that 
curve. 
Let O be the lowest point of a uniform string AOB, suspended at the points A and 
B, and let the tension at O be denoted by r, and at any other point C by T. Then if 
we consider the equilibrium of the portion 
O C we find that the forces acting on it are 
r, T, and its own weight W ; and these are 
parallel respectively to the sides of the tri- 
angle C D E. Hence, by the property of 
the triangle of forces, 
W 
— =tan E CD= tan <p, 
where <p is the angle which the tangent at 
C makes with the tangent at O. Now if s 
be the length of O C and c the length of a 
portion whose weight is equal to t, we have, since the string is uniform, 
s_W 
c r ’ 
.-. s=ctan<p (80) 
34. The equation s=ctan<p, which we have just obtained, is the intrinsic equation 
of the catenary ; we get the Cartesian equation from it as follows : — Make OF=c, and 
draw FX parallel to CD. Then we shall take these lines as axes. Now let the coor- 
dinates of the point C be denoted by x and y, and we have 
dy 
^=sin <p ; but ds=c sec 2 <p d<p, equation (80), 
Again, we have 
and 
Hence 
:.y=c sec<p. 
dx 
^= c °s<P, 
dx=c sec <f> dq, 
;r=<?log(sec <p-f tan <p), 
f=b c = e l sec<p. 
y = o{e c +e ~ e ) ; 
( 31 ) 
and this is the Cartesian equation of the catenary. 
35. From the value 
x=c log (sec <p + tan <p) 
