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P, at A may be replaced by 2P, at 0 and P, at D. Pat B may be replaced by 2P, at O and P, at C: 2P, and 2P, at O counteract one another, so that we are left with P1 at C and P, at D, as equivalent to the original couple. But these two forces constitute a couple like to the original one, equal to it and in the given plane parallel to it: therefore as the original couple is equal to one couple in the parallel plane, it is by (i) equal to any like couple of the same moment in that plane.

64.* The latter part of the last proposition might have been proved in a manner analogous to that adopted for the former, as follows.

Let A and B be the two couples: we shall prove that A and B reversed satisfy the sufficient conditions of equilibrium of Art. 55.

Take three straight lines, intersecting in a point, one perpendicular to the plane of each couple, and the other two in the plane of B.

It is obvious that the algebraical sum of the resolved parts of the four forces in each of these directions is zero: also the moments of A and B reversed, about the line perpendicular to their planes, are numerically equal but of opposite sign. Hence the algebraical sum of the moments of the four forces forming them about this line is zero. The moment of each of the forces forming B reversed about any line in their plane is zero, and the moments of the two forces forming A, about any line in the plane of B, are equal numerically but of opposite sign; the algebraical sum of the moments of all four forces about every straight line in the plane of B is therefore zero.

The six sufficient conditions of equilibrium of Art. 55 are therefore satisfied, and the couples A and B reversed balance one another; in other words A and B are equivalent.

Ex. 1. Like parallel forces, each equal to P, act at three of the corners of a rhombus, perpendicular to its plane: at the other corner such a force acts that the four forces are equivalent to a couple : find the moment of the couple, provided the angle of the rhombus at which the last force acts is 60o. Ans. 2√3. Pa, where a is a side of the rhombus.

Ex. 2. ABCDEF is a regular hexagon: equal forces act along AB, BC, DE, EF, and two other forces, each double any one of the former forces, act along DC and AF: prove that they maintain equilibrium.

65.* Let us consider what we require to know to determine the effect of a couple on a rigid body. It is unnecessary to know the actual position of the plane in which the couple acts, but we must know the direction of the plane, i.e. the direction of a line to which it is perpendicular. We do not require to know the magnitude or direction of the forces which compose the couple, but we must know the magnitude of its moment and its sign, i.e. the direction in which it would tend to turn the body round a line perpendicular to its plane, the line being fixed and the body rigidly connected with it.

Now a straight line at right angles to the plane of the couple, and of length proportional to the magnitude of its moment, will represent the couple in the first two respects: also, if it be understood that the line is drawn in that direction in which the axis of a right-handed screw moves, when it rotates in the same way as the couple tends to turn the body, the sign of the couple will also be represented.

In fig. 39, if the arrowhead on the circle indicates the direction in which the couple would tend to turn the body about AB, supposing the

latter fixed and the body rigidly connected with it, the sign of the couple in accordance with the above convention would be represented by AB and not by BA.

The line which thus completely represents the couple is termed the axis of the couple.

66.* We shall now prove that couples follow the Parallelogram Law, in other words, that

If from a point the axes representing two couples be drawn, and a parallelogram be constructed on these two axes as adjacent sides, the diagonal passing through the above-mentioned point is the axis of a couple equivalent to the two, i.e. of their resultant couple.

We may suppose the couples to consist of forces acting at the ends of a common arm, in which case the

Fig.40

moments of the couples will be respectively proportional to the forces composing them.

Let Aa be the common arm, and let AB, ab represent the two equal and parallel forces forming the first couple, AC, ac those forming the second.

Draw AB' perpendicular to Aa and AB, equal to AB, and in the direction which by the convention of Art. 65 represents the sign of the first couple: similarly draw AC' perpendicular to Aa and AC, equal to AC and in the proper direction. Then AB' and AC' are the axes of the

two couples.

Complete the three parallelograms, ABCD, abcd, AB'C'D, and join AD, ad, AD'. These parallelograms are clearly equal in every respect, so that AD = ad = AD'. Also AD, ad are parallel, and AD' is perpendicular to AD.

But the two forces AB, AC are equivalent to AD, and the two ab, ac, to ad, so that the two couples are equivalent to AD, ad, which form a couple of which AD' is the axis. Hence the couples whose axes are AB', AC are equivalent to a resultant couple of which AD' is the axis.

Cor. Hence we may deduce propositions relating to the composition and resolution of couples, analogous to those obtained in Arts. 19-26, 30-32, relating to the composition and resolution of forces.

67.* Prop. Any system of forces acting on a rigid body can be reduced to a single force acting at any arbitrarily chosen point and a couple.

Let A be the arbitrarily chosen point, P any one of the forces.

Fig.41

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We shall not alter the effect of the forces by applying at A two forces P,, P, each equal and parallel to P, and in opposite directions to one another. P, which is opposite to P, forms with P a couple. Hence P is equivalent to P1 at A, and a couple.

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The couple vanishes in the case in which A lies in P's line of action.

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Similarly we may replace each of the other forces by a force at A, equal to it and in the same direction, and a couple.

The whole system thus reduces to a series of forces at A, respectively equal to and in the same direction as the several original forces, and a series of couples. But the forces at A are equivalent to a single resultant at A, and the couples to a single resultant couple.

Cor. The magnitude and direction of the single resultant is the same wherever A is, and the resultant couple is the same for all positions of A in a line parallel to the single resultant force.

68.* Prop. Any system of forces acting on a rigid body is equivalent to a single force and a couple whose axis is parallel to the direction of the single force.

By the last proposition, the system is equivalent to a single force R acting at any given point A, and a couple H. If the axis of H make an angle & with the direction of R, it may be resolved into H cos & in the direction of R and I sin at right angles to that direction.