177
TRACT Iir.
Formulae for determining any four of the six quantities <p, c, t, x, y, and z in a catenary, whose thickness (that is, the area of its section,)
is at every point as the tension at that point, the other two being given.
Let the six quantities be denoted as follows: <p = the angle; c — the constant quantity or tension at the apex; t — the tension at <f>; z — the length of the chain between the pointof suspension at $ and the apex; x — the absciss; y — the ordinate; and m — 2.302585, the number by which the common logarithm of any number must be multiplied to ob-tain the hyperbolic logarithm of the same number; <1 — al 1 4 8 1 ^ - — .0174533 = the length of an arc of one degree in a circle whose diameter is 2, or radius 1; if — the number incolumn y opposite the angle $ in the catenarian table; and tp° — the number of degrees in the angle <p.
Given
Quan-
tities.
REQUIRED QUANTITIES.
c and pp and tp andp and xp and^c and tc and zc and xc andyt andt and xt and y'z and xz andy
x and^
sec ,p
y’ =
log.sec.p =
cm
C.
^ cdt sec.p
z ~ y<
x log.sec.p X
mt ~~ sec.pdt sec.p J
y ~ f
z __ log.tang.(45°-Hf) X
x — iog.sec.p
dz log.t ang.(45° + ^) %
dx log. sec.p X
my p°
t cos. p
y
m log.sec .53
y
d p°
sec.i
t cos .pz
y’
z
y’
y
d p°
t.
c sec.p
z sec.p
y'
x sec.p
m log.sec.iy sec.pd p°
c sec.pc sec.pc sec.p
z sec.^i
y’
z sec.p
y’
y sec.pd p°
■ ty’
°y
— t y’ cos.p
_ ±y_ _
m log.sec.^
y y'
d p°c y’
cy' *cy' *
cos.f. t y't y' cos ,p
yy
y •
t d p°
sec.p
c d p°
— t d p°.cos.pzd p°
y
x d p°
m log.sec.p
cd p°c d p°cd p°
z d <p°
y
t cos.p d p°
z d p°
y'
x.
c m log.sec.c/,trn log.sec.c^
— tm cos.<£.log.see p
sec.ip
m z log.sec.t/)
y'
my log.sec.<£
d <t>°t
c m log. — — cm log.sec.^c m log.sec.f
c m log.sec.i^mz log.sec.^>
y'
t m cos.<p.\og.sec.<j>
m z log.sec.<£
y'
* y' may be found, p being obtained in the column or y being known p may be found by the table.+ Which quotient found in column—in the table, p and y both become known,t From which p may be found by approximation, and from p y becomes known.
THE END.