Cook.—On the Construction of the Brunner G orge Dridge. 313 
Equations (2) and (3), therefore, become, 
d rp dy w dac du: ~ 
oTa) gno Seo . (5) 
ü f, ds w dm i H 
ae (0 3) — = sa eee 
The latter of these equations integrates at once, and gives us,— 
r£ -q-i 
but, when x = 0 we have i = 0, therefore k = 0 
hence T a _ wr 
ds 2 
But from (4) TE ae 
Dividing one equation by the other,— 
dz w 
pi mA rr - v 
da do- 
Integrating, and observing that x and + vanish together, we get, — 
xpo as x? £s (7) 
or, Ga 4 2 2 à (8) 
. n 
This is the equation to a parabola, and gives the parabola in which it is 
well known the chain would hang if everything were in a vertical plane. 
To integrate (5), we observe that tan 0 =, substituting this value we 
obtain,— 
ds 
, m ay mn dy | dz dy 
e Ta ip aa t 
and from (7) X o Xy 
z -2 
r Wx 
: i ES ANY Sag de. 
hence we obtain c -i ($4) — 9c Fao 
2 
: Pall S 
or oh e Qy = 0 
This can be reduced to a linear equation by the well-known substitution 
. of putting v = e ?, and changing the independent variable to 9. In this 
way the complete integral will be if a and b are the constants of inte- 
gration, 
y = av? + i 
And since v and y vanish together, b must be zero, hence the last 
equation reduces to 
y = ax? js (9) 
This equation together with (8) determines the form of the curve in 
which the chain hangs. The constants a and c, Which enter into them, 
are easily determined from the condition that the chain passes through the 
top of the pier, whose co-ordinates are, adopting a foot as our unit of length, 
æ = 150, y = 11,2 = 25; and w = a = tons 
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