the reaction between the upper and any of the lower spheres. From the equilibrium of the upper sphere, resolving vertically,

The resultant of W acting vertically, and along OA, on the sphere

W 2√2

whose centre is A, makes with the vertical the angle tan

i.e. tan-1 },

But this resultant is equal and opposite to the pressure of the bow which acts along AO'.

But the radius of the hollow sphere is equal to O'A together with r, therefore radius of the bowl (213 + 1) r.

If the bowl is any larger, O' will be further from H, and for the pressure of the bowl to counteract the resultant of the other forces on the sphere (centre A), we shall have to suppose that the actions of the two adjacent lower spheres on it are towards their respective centres instead of away from them. But as the spheres are incapable of exerting such forces, equilibrium is not possible, i.e. the spheres will separate.

Ex. 8. A heavy bar, AB, is suspended by two equal strings of length 1, which are originally parallel: find the couple which must be applied to

W

the bar to keep it at rest after it has been twisted through an angle in a horizontal plane.

Let C, D be the fixed ends of the strings; CA', DB' the original vertical positions of the strings.

Draw Aa, Bb at right angles to CA', DB' respectively. Join ab cutting AB in its middle point G. Let 2a be the length of AB, and 4=angle aCA or bDB.

Then

CA sin a CA aA=2AG sin haGA,

=

.. l sin p=2a sin 10....

..(1).

Let T be the tension of either string: they will from symmetry be the same.

Let P be the magnitude of the force which applied horizontally in opposite directions at A and B, at right angles to AB, will keep the rod in equilibrium.

Resolving vertically, we have

W-2T cos p=0.

Taking moments about the line of action of W, we have

Hence

2P. a -2T sin. a cos 30=0.

2Pa=

a W sin cos 0

cos

(1) and (2) enable us to determine 2Pa in terms of a, l, W and 0.

.(2).

In this example we have assumed as obvious that a couple only is required to maintain equilibrium: it can be shewn however that the values we have obtained for P and T will satisfy the six conditions of equilibrium of Art. 55.

EXAMPLES.

1. Four points A, B, C, D lie on a circle and forces act along the chords AB, BC, CD, DA, each force being inversely proportional to the corresponding chord: prove that the resultant passes through the common points of (1) AD, BC; (2) AB, DC; (3) tangents at B, D, and (4) tangents at A and C.

2. If six forces acting on a body be completely represented, three by the sides of a triangle taken in order, and three by the sides of the triangle formed by joining the middle points of the sides of the original triangle, prove that they will be in equilibrium if the parallel forces act

in the same direction, and the scale on which the first three forces are represented be four times as large as that on which the last three are represented.

3. Forces P, Q, R act along the sides of a triangle ABC, and their resultant passes through the centres of the inscribed and circumscribed circles: prove that

4. Prove that a uniform rod cannot rest entirely within a smooth hemispherical bowl, except in a horizontal position.

5. If a uniform heavy rod be supported by a string fastened at its ends, and passing over a smooth peg; prove that it can only rest in a horizontal or vertical position.

6. A heavy equilateral triangle hung upon a smooth peg by a string, the ends of which are attached to two of its angular points, rests with one of its sides vertical; shew that the length of the string is double the altitude of the triangle.

7. A fine string ACBD tied to the end A of a uniform rod AB of weight W, passes through a fixed ring at C, and also through a ring at the end B of the rod, the free end of the string supporting a weight P; if the system be in equilibrium prove that AC: BC :: 2P+W : W.

8. A horizontal rod, the ends of which are on two inclined planes, is in equilibrium: if a, ẞ be the inclinations of the planes, prove that the centre of gravity of the rod divides it into two parts in the ratio of tan a to tan ß.

9. A uniform heavy rod AB has the end A in contact with a smooth vertical wall, and one end of a string is fastened to the rod at a point C such that AC=AB, and the other end of the string is fastened to the wall; find the length of the string if the rod is in equilibrium in a position inclined to the vertical.

10. A cylindrical ruler whose radius is a, and length 2h rests on a horizontal rail with one end pressing against a smooth vertical wall, to which the rail is parallel. Shew that the angle the axis of the ruler makes with the vertical is given by (h sin 0+a cos 0) sin2 + 2a cos 0=b, where b is the distance of the rail from the wall.

11. Two equal heavy spheres of one inch radius are in equilibrium within a smooth spherical cup of three inches radius. Shew that the

pressure between the cup and one of the spheres is double the pressure between the two spheres.

12. Along each side taken in order of a polygon inscribed in a circle, acts a force whose magnitude is proportional to the sum of the lengths of the two adjacent sides: prove that the system of forces is equivalent to a system of forces acting along the tangents at the corners of the polygon, each such force being proportional to the length of the chord joining the two adjacent points.

13. ABCD is a quadrilateral: forces act along the sides AB, BC, CD, DA measured by a, ß, y, ô times those sides respectively. Shew that if there is equilibrium αγ = βδο

Shew also that AABD|AABC= a (y − ẞ)/d (ẞ − a).

14. Into the top of a fixed smooth sphere of radius a is fitted firmly a fine smooth vertical rod. A bar of length 2b has at one end a ring which slides on the rod; and the bar rests on the sphere. Shew that in equilibrium the angle (a) the bar makes with the horizontal is given by

a sin a=b cos3 a.

15. Forces P, Q, R act along the sides BC, CA, AB of a triangle; shew that their resultant will act along the line joining the centre of the circumscribing circle with the orthocentre if

16. A kite (weight P) having a tail (weight Q) is stationary, with a normal to its face, the direction of the wind, which is horizontal, and the string in the same vertical plane. The tail is attached at a distance a below the kite's centre of gravity, the string at a distance b above. Shew that, neglecting the action of the wind on the tail, the inclination of the kite to the horizon is given by the equation

IIb sin2 0 = {Pb+Q (a+b)} cos 0,

where II is the pressure on the kite, when placed perpendicular to the wind's direction.

17. Forces act at the middle points of the sides of a rigid polygon in the plane of the polygon; the forces act at right angles to the sides, and are respectively proportional to the sides in magnitude: shew that the forces will be in equilibrium if they all act inwards or all act outwards.

18. Shew that it is impossible to arrange six forces along the edges of a tetrahedron so as to form a system in equilibrium.

19. On the sides of a right-angled triangle ABC squares are described, the square BCDE on the hypotenuse on the same side of BC as A, and the squares CAFG, ABHK on CA, AB on the opposite side of each to the triangle: prove that the forces represented by the straight lines AB, BC, CA, BH, HK, KA, CD, DE, EB, AF, FG, GC will form a system in equilibrium.

20. If four parallel forces balance each other, let their lines of action be intersected by a plane, and let the four points of intersection be joined by six straight lines so as to form four triangles; then prove that each force is proportional to the area of the triangle whose angles are in the lines of action of the other three.

21. Two rings of weight P and Q respectively, slide on a string, whose ends are fastened to the extremities of a straight rod inclined at an angle to the horizon: on the rod slides a light ring through which the string passes so that the heavy rings are on different sides of the light ring. Prove that in the position of equilibrium the inclination of those parts of the string next the weightless ring, to the rod, is given by the equation tan p/tan 0= (P+Q)/(P~ Q).

22. An elastic string passes round three equal right-circular cylinders so that when each cylinder touches the other two along a generating line, the string is just not stretched: shew that if the system be placed on a smooth horizontal plane, the inclination (0) of the plane containing the axis of the upper cylinder, and that of either of the lower ones to the horizontal, in the position of equilibrium, is given by the equation (+3) W=2x (2 cos 0-1) tan 0. (W is the weight of the upper cylinder, and X is the modulus of elasticity.)

23. Two equal circular discs, of radius r, with smooth edges are placed on their flat sides in the corner between two smooth vertical planes inclined at an angle 2a and touch each other in the line bisecting the angle; the radius of the least disc which may be pressed between them without causing them to separate = (1-cos a)/cos a.

24. A rectangular lamina ABCD is supported with its plane vertical and one edge AB in contact with a smooth vertical wall, by an endless string which passes through smooth rings, one fixed to the wall at A, and two others P, Q fixed in the sides AB, CD of the lamina respectively