Mr. Davies Gilbert on the mathematical theory , &c. 203 
the formulas from which my approximations were derived ; 
adding to them other formulae and tables for the catenary of 
equal strength. A curve not merely of speculative curiosity, 
but of practical use, where a wide horizontal extent may 
chance to be combined with natural facilities for obtaining a 
correspondent height for the attachments. 
Both the ordinary catenaries, and these of equal strength, 
like circles, parabolas, logarithmic curves, &c. have the pro- 
perty of being each identical with themselves in every respect 
but size : and as the radius, the parameter, and the sub- 
tangent give the respective magnitudes of these curves, so 
are the catenaries determined in magnitude by the tension 
(expressed in measures of the chain), which takes place at 
the middle point, or apex of the curve, where it is a minimum. 
Consequently, when this tension is determined or given, all 
the other relations may be expressed in the same manner as 
sines, cosines, &c. in the circle. 
I assume that the first principles of the catenary curve 
are known ; they will, consequently, be noted with no other 
view, than to derive from them ulterior properties. 
For the ordinary catenary : 
Let a = the tension at the apex, estimated in measures of 
the chain ; 
x = the absciss, the versed sine, or depth of curvature ; 
y = the ordinate, or semi-transverse length ; 
z = the length of the curve. 
Then since the tension, (a,) acts horizontally at the apex (A,) 
since the weight of the chain (z) acts at right angles to the 
former, and the force of suspension at (P) acts in the direction 
of the tangent. These forces must be represented in direc- 
