124

STATIC'S.

Pkob. 2. A cord AA t A 2 ...a is held at rest by forcesacting at its extremities and at the knots A x A 2 A s ...in givendirections: having given the form of the polygonal figure ofthe cord required to find the relations of the forces; also tofind the tensions of the portions of cord: fig. 6l.

The portions of cord need not be in the same plane; butthe force which acts at any knot, as Pi at A x , must have itsdirection in the plane of the portions of cord which join in A l .Let P, Pa...be the forces acting at the knots A X A 2 ... : T T x P 2...P„ the tensions of the portions of cord: a x ^i, a 2 /3 2 ,„.theangles which the directions of P]P 2 ...make respectively withthe portions of cord at the knots.

Then A l is held at rest by the three forces P X T X T-,hence, resolving these forces in the direction of P l and at right

angles to this, we have by Art. 23.

P,— T cos — Pj cos j3i =0. (l),

T sin on— Pi sin /3j = 0. (2).

Again, A 2 is held at rest by Pj P s P 2 ; hence

P 2 —Pi cos a 2 — P 2 cos jS 2 = 0 . (3),

Pi sin o 2 - P 2 sin /3 2 = 0. ( 4 ),

and so on: if there be n knots we shall have 9,n equations,

involving 2w + 1 unknown forces P X P 2 ... P n T T x .P„:

we shall therefore have an equation of condition connectingthese forces, we shall suppose P to be known.

By equations (2) (4) .we have

P

T \ = -- n

sin p

sin /3, P, sin/3 2 T„_ x sm f.

sin en sin on sin

T, T 2 = — w . n. T, and so on,sm fi 1 sm p 3

and the tensions are all known in terms of P.

Also by (1) (2) eliminating P,,

p T sin &

1 sin («i + /3,)

, and in like manner