By substituting from the other equations in (1) and (9), we have

Ex. 4. A gipsy's tripod consists of three uniform straight sticks freely hinged together at one end. From this common end hangs the kettle. The other ends of the sticks rest on a smooth horizontal plane, and are prevented from slipping by a smooth circular hoop which encloses them and is fixed to the plane. Shew that there cannot be equilibrium unless the sticks be of equal length; and if the weights of the sticks be given (equal or unequal) the bending moment of each will be greatest at its middle point, will be independent of its length, and will not be increased on increasing the weight of the kettle.

Let OA, OB, OC be the three rods, P, Q, R their respective weights acting at their middle points. Let X, Y, Z be the vertical stresses at A, B and C, and X', Y', Z' the horizontal stresses.

Draw OH vertically downwards.

The three forces acting on OA, viz. P and the resultant stresses at and A, must be in one plane (Art. 61) the vertical plane containing 04, i.e. OAH.

X' the horizontal stress at A must therefore act along AH; similarly Y' and Z' act along BH and CH respectively.

But these horizontal stresses act along the normals to the circle ABC, so that I must be the centre of that circle. The lines HA, HB, HƠ

must therefore be equal to one another, and also OA, OB, OC to one another.

Let 21 be the length of each rod, its inclination to the horizon.
Taking moments about O for the equilibrium of OA, we have

X.21 cos - Pl cos 0- X'. 2l sin 0 = 0

.. 2X - P-2X'tan 0=0.

The bending moment at a point on OA distant x from A

This is clearly a maximum, when xl, i.e. the bending moment is greatest at the middle point, where it is equal to Pl cos 0, or Pr, where r is the radius of the hoop, i.e. is independent of 1 and W.

Ex. 5. An elastic band binds together any number of smooth right cylinders so that each cylinder touches only two others. Prove that if lines be drawn from a point parallel and proportional to the pressures between the cylinders, their extremities will lie on a circle.

Let Aa, Bb, Cc, &c. be the portions of the band in contact with the cylinders A', B', C', &c.

From any point O draw a number of equal straight lines Oa, Oß, Oy, Os, &c. respectively parallel to the portions of the band zA, aB, bC, cD, &c. These lines will therefore represent the tensions along the corresponding portions of the band.

Join aß, ẞy, yd, &c.

By the triangle of forces, aß represents the resultant action of Aa on the cylinder A'. Similarly ßy, yd, &c. represent the resultant actions of the band on the cylinders B', C', &c. respectively.

Through ẞ draw ẞO' parallel to the normal common to A' and B': through y draw yo' parallel to the normal common to B' and C'. Join d0', €O', &c. By the triangle of forces O'ẞ and yo' represent the pressures of the cylinders A' and C' on B'.

Therefore O'y, yd represent two of the forces on the cylinder C', so that 80' must represent the third, which is the pressure due to D'. Similarly it can be shewn that O'a, &c. represent the pressures between the other pairs of cylinders.

Hence from O' straight lines O'a, O'ß, O'y, &c. have been drawn representing in magnitude and direction the pressures between the cylinders, and their extremities a, ß, y lie on a circle whose centre is 0, since Oa, Oẞ, Oy, &c. are all equal.

(The cylinders are not necessarily circular.)

EXAMPLES.

1. Two uniform heavy rods, each of length a and jointed together by a smooth hinge, are placed symmetrically over two pegs at a given distance b apart in a horizontal line; prove that in the position of equilibrium each rod is inclined to the horizon at an angle cos-1 (b/a)3.

2. Three equal uniform rods, AB, BC, CD, of the same material and thickness, are jointed at B and C. If they are supported in a horizontal plane by smooth pegs placed under AB and CD, shew that the distance between either peg and the nearest joint is one-third the length of a rod.

3. A uniform heavy rod of length 2b and weight W can turn freely about one end. To this end is attached a string of length 1(<2b), which supports a sphere of radius a and weight W'. When the system is in equilibrium with the rod resting against the sphere, the rod makes an angle with the horizontal; shew that tan 6-tan a=Wb/W'a, where l=a (sec a-1), and 12+2al is <4b2.

4. A uniform heavy rod hangs by light inextensible strings, attached to its ends, and also to the ends of another uniform rod, which can turn about a pivot at its middle point. Prove that, when there is equilibrium, either the rods or the strings are parallel.

5. Prove that the angular points of a funicular polygon, in which the weights are equal and also the horizontal distances between them, lie on a parabola.

6. Two rods AC, BC, of equal uniform thickness are jointed at C, and the ends A and B are fixed at two points in the same vertical line. Prove that the direction of the action at the joint C bisects the angle ACB: and if AB2=4AC.BC, shew that its magnitude is equal to a quarter of the difference of the weights of the rods.

7. A chain formed of rods of equal weight jointed together is hung up by its two ends and rests under the action of gravity. Shew that, if lines be drawn from a point representing the actions at the hinges, their ends lie on a straight line.

8. A rhombus is formed of four similar uniform rods connected by smooth hinges at their extremities, and two of these rods rest upon two smooth pegs in the same horizontal line: determine the position in which the rhombus will rest with one of its diagonals vertical.

9. Two uniform rods AB, AC, of lengths a, b respectively, are of the same material and thickness and smoothly jointed at A. A rigid weightless rod of length l is jointed at B to AB and its other end D is fastened to a smooth ring sliding on AC. The system is hung over a smooth peg at A: shew that AC makes with the vertical an angle

10. A regular tetrahedron consists of six rigid bars without weight. It is suspended from one angular point, and from the other three equal weights W are hung: find the strain on each of the horizontal edges.

11. A beam AB of length a and weight w rests horizontally on two smooth pegs, whose distances from A and B respectively are ja and a: if from A a weight 5w is hung, and from B, w, shew that the bending moment is greatest at the peg next A, and find its magnitude.

12. Two heavy uniform rods AB, BC, weights P and Q, are connected by a smooth joint at B. The ends A and C slide by means of small smooth rings on two fixed rods each inclined at an angle a to the horizon. If 0 and ☀ be inclinations of the rods AB, BC respectively to the horizon, shew that