10
THE PRINCIPLES OP BRIDGES.
TRACT I.
PROP. II.
If any number of lines, that are connected together and
moveable about the points of connection, be kept in equilibria by
weights laid on the angles, as in the last proposition: Then
•wil-l the weight on any angle c be universally proportional to
sine of the Z bcd . . , . ,
- -; that is, directly as the sine of tnat
S. Z EC C X S. Z. CCD J
angle, and reciprocally as the sines of the two parts or anglesinto which that angle is divided by a line drawn through itperpendicular to the horizon. See the former figure.
Demonstration .—By the last proposition the weights are.as b b, c c, T>d, &c, where Bn = pc, c q — rD, ds = t: e, &c.But,, since the angle ab b is = the angle b bn, and the anglebcc = the angle ccq, &c, these being always the alternateangles made by a line cutting two other parallel lines; alsothe sine of the Z abc = s. Z b nb, and s. Z bcd = s. Z cqc,these being supplements to each other; by plane trigonometrywe shall have,
(bw = )
(c q~)
( m = )and so on.
B b X
s.
Z. AB b
(c P=)
CC
X
s.
ZL. CCD
s.
X.
ABC “
s.
z.
BCD ’
CC X
s.
Z. BCC
(Dr = )
vd
X
s.
z. c/de
s.
z.
BCD ~
s.
z
CDE ’
T)d X
s
Z. CD d
w
11
ve
X
s.
Z CEF
s.
z.
CDE
S.
z.
DEF ’
Hence,%b : cc : :
cc :vd : :
Txl: Be : :
s.
z
ABC
S.
BCD
s.
z
AB b
s. z
ccd’
s.
z
BCD
s. Z
CDE
s.
z
BCC
S. Z
c/be’
s.
z
CDE
S. /-
DEF
s.
z
CD d '
S. zL
cef’
-, &c.
Or, by dividing the latter terms of the first of these pro-portions each by s. Z bsc, and then compounding togethertwo of the proportions, then three of them, &c, striking outthe common factors, and observing that the s. Z bsc is ==
/