CASE II. Figure 11.—When the point F, is taken in one of the extreme lines of the direction, A D. On.D F, as a diagonal, complete the parallelogram D Ere; draw F II per pendicular to D B, and F I to D c; then because G F H and E F I are right angles, and the angle I F 1I is common, the remaining angles I F G and a F E are equal ; and because the angles F I a and F n E are right angles, the triangles G I F and Ear are similar ; therefore FG:FE::FI:FII; that is, the forces at B and c are reciprocally as the distances of their lines of direction taken from any point in the lines of direction oldie other force. Therefore, universally, if any three forces be in cquilibrio, any two of them are recipro cally as the distances of their lines of direction, taken from any point in the line of direction of the other force.
Proposition X. Figures 12 and 13.-11 a solid, A B c, be supported by three forces in the lines of direction z B, A W, c J, these three lines will have the same point of concourse, or be parallel to each other.
Join A B, B C, c A; 110W the triangle A B C cannot be in cquilibrio, unless the directions have the same point of con course, or be parallel, and also in the plane of the triangle : otherwise the threes at the two points in each of the sides of the triangle would not be equal in opposite directions ; there fore the equilibrium of the triangle would be destroyed, as has been shown by Proposition V111., and consequently that of the solid would also be destroyed.
Corollary 1.—The intensities of any three forces keeping a solid in epuilibrio, will be as the parts of a parallelogram formed on their lines of direction.
Corollary 2.—Likewise the intensities of any two forces are reciprocally as the distances of their lines of direction from any point in the line of the other force, whether their directions meet or are parallel to each other.
Corollary 3.—Ilence, in three forces acting upon a pris matic solid, or lever, in parallel directions, any two forces will be to each other in the reciprocal ratio of the distances of their lines of direction, on the opposite side of the solid, to the direction of the other force.
Proposition X1.—If' a solid be in equilibrio by three forces, and if any point be taken in the line of direction of any one of them : the products of each of the other two, by the dis tance of their respective lines of direction front that point, will be equal. Figure 14.
Let A, 13, c, represent the intensities at A, n, c, in the directions E A, 0 B, F C. Take any point, n, in the middle line of direction, and draw D E, D F perpendicular to the other two lines of direction ; then (Corollary 2, last Problem) A:C::DF: DE; therefore ADE=CXDF. Again, from any point, F, in one of the extreme lines of direction, draw G and F Il perpendicular to the other two, then A: 11: : F 0 : 1 II; therefore A X F H= B XI.' G.
Corollary 4.—Hence, if three forces act perpendicular to a prismatic rod, or beam, the products of any two, each by its distance on the beam from the third, are equal.
Corollary 5.--Hence, if three forces act perpendicular to a prismatic rod, or beam, the products of any two of their distances 1i•om the third, in the direction of the beam, are equal ; for in this case all the lines, F D, D E, F 0, F H, coin cide in one straight line, and become parallel to the beam, and the segments intercepted by the directions are equal to those on the beam.