314 Transactions.— Miscellaneous. 
2 
R 11 H 41 (150). za 123 = : 
.ü = 50)? c= ub ig x 150 x ^95 oni uie tension 
at lowest point in tons. 
From equations (8) and (9) we deduce 
4 _*,%=0 io: 3d 
the equation to a plane passing through the axis of x, that is, the straight 
line joining the feet of the suspending rods. Considering the curve in 
which the chain hangs, as determined by equation (10) combined with 
either (8) or (9), we see that the curve is that made by the intersection of 
the plane (10), with either of the parabolie cylinders (8) or (9), and hence 
is not only a plane curve, but is a common parabola. 
Next to calculate the tension at any point. We have already shown 
that 
E ae == ¢ te T = = 
(2) = 14 (2) (E) 
and Au = %ax = i50? 
dz w 
T EE 2 
um 3X 150 
By making the necessary substitution it is then easy to find the tension 
at any point of the chain. At the point where the chain passes over the 
top of the pier « = 150, hence then a = E 2 = " 
Therefore the tension at that point is 
eV 14 ba. = = Vw? +257 + 1a = = Vam = 65°45 tons nearly. 
The portion of the chain between the top of the pier and the anchor- 
plate will hang by its weight in a catenary, and the tension at the 
anchor-plate would be less than that at the top of the pier, by the weight 
of a piece of the chain equal in length to the vertical height between the 
two points. But, as we have neglected the weight of the chain all along, 
we must consider the tension on the anchor-plate to be equal to that at the 
top of the pier, viz., 65:45 tons. 
If the chain and suspending rods had all lain in a vertical plane, equa- 
tions (1) and (8) of our fundamental equations would have applied, and the 
chain would have then hung in the parabola whose equation is (8), and the 
c, Which is a eonstant introduced by integration, would be the same as in 
the other ease. It can be proved, as in the previous case, that the tension 
on the rope at the top of the pier would be 
evi t= x v= V 
The ratio of the tension in the construction actually adapted to the tension 
which would have existed had the whole been in a vertical plane, is / $371, 
