1 58 
proposed to be erected at Montrose . 
higher ; but I have thought proper to take the lowest estimate. 
Should I, however, be entrusted with the erection of the proposed 
chains ; and it is an exact parabola again, when the weight of the chains is as 
nothing compared with that of the roadway ; and the latter is in general by 
much the nearest to the true state of the case. In the bridge above propo- 
sed, the weight of the roadway itself is four times that of the chains, and, 
when loaded to its utmost, it is ten times greater. This consideration is of 
importance in practice, much embarrassment having been felt in loading the 
bridge with its roadway, by the weight gradually altering the figure of the 
curve ; and this will always take place if the lengths of the suspending rods 
be drawn to the curve of the catenary. As the progress of loading proceeds, 
the figure of the arch will change, the roadway will be drawn off its level, the 
suspending rods off the perpendicular, and the whole structure will be dis- 
torted, besides that different parts of it will be strained beyond what they 
are intended to bear. To avoid such evils, the length of the suspending 
rods must be calculated by that figure to which the weight of the road-way 
will finally bring the chains ; and, for this purpose, the curve of permanent 
equilibrium cannot be too nicely investigated. But for calculating the least 
and greatest strains in the arch, we may, without sensible error, assume it as 
a parabola. Now, in all these funicular curves, whether catenary, parabola, 
or any other figure, it is a general property that the horizontal tension is 
proportional to the radius of curvature at the vertex ; and is, in every case, 
equal to the weight of as much of the curve as is equal to the radius of cur- 
vature in length, and of the same thickness or cross section as the curve at 
the vertex. But the radius of curvature of the parabola at the vertex, is just 
half the parameter ; and hence, from the well known equation of this curve 
y 2 — px , we deduce T, the horizontal tension T = \p = — . But y is half the 
c V 
span of the arch = |S, S denoting the span, and x is its depth or versed sine 
— d ; hence T = — — . Let d now be = — , and S — nd, then T — 
ttd n r>d a 
But S denotes the weight of a part of the bridge, equal to the span in length, 
and having every where the same cross section as the chains and roadway in 
the middle ; and this is evidently within a mere trifle of the whole weight of the 
bridge, only falling short of it by the weight of a part of the chains, equal in 
length to the difference between the arch and the span, and which will not, 
in general, amount to the 200th or 100th part of the whole weight. For S. 
therefore, we may safely substitute W, and this gives the formula — , al- 
8 
ready stated. 
The same rule may easily be deduced by the principles of fluxions. It 
is evident that the fluxion of the ordinate, is to that of the abscissa as ra- 
dius to the tangent of the angle which the curve makes with the ordinate, 
and which is termed the Angle of Deflection ; that is, dy : dx : : R : Tang. D. 
But as the weight of each half of the bridge, according to what we have seen, 
is equal to the ordinate y ; and as this weight is also in every funicular curve 
