a length) or its direction (given by an ort), so it is sometimes convenient to speak of the vector part, or simply the vector, of the Rotor.

୪

α

The vector of a rotor will, like any other vector, be denoted by a small Greek letter a, B,... The rotor itself will then be called the rotor a, the rotor B,... This will not lead to any confusion, if the line of position is properly indicated. A simple way of giving the rotor is by its vector a, and the position vector of some point in the axis. Thus in the figure O is a known origin, QQ the axis of the rotor, p the position vector of P any point in QQ and a the

e

Fig. 100.

α

vector of the rotor.

The rotor a is completely specified when the vector a and the position vector p are known.

152. Two rotors are said to be equal if they have the same vector and the same axis.

Two rotors are said to be equal and opposite if they only differ in sense.

153. We proceed to define the laws for the Addition of Rotors.

A Rotor, whose magnitude is a length, is a purely geometrical quantity and for such Rotors the laws might be chosen quite arbitrarily. As however certain physical quantities can be represented by Geometrical Rotors (the magnitudes being represented to scale by lengths), it is desirable to choose the laws so that they will follow the combination of those things represented by the Rotors.

As a guide we will take the well-known laws for the combination of forces acting on a rigid body. By this means we shall assure that the results obtained for Geometrical Rotors will hold for forces acting on a rigid

body. Whether they will hold also for rotations for finite angles and for spins would require separate investigation. As a matter of fact they hold for spins but not for finite rotations.

After the Geometrical Theory of Rotors has been developed we shall shew (§ 179) that the theory applies to Forces. Meanwhile it may be stated that a Force is an example of a Rotor Quantity, and that the well-known phrase, Moment of a Force, corresponds to our Momental Area of a Rotor. The word Rotor is used instead of Force because the theory applies to other quantities besides forces.

B

A

154. Momental Areas of Rotors.

Suppose xx the axis of a rotor and a its vector, and

Fig. 101.

α

let P be the position of some point A in the axis with reference to an origin 0.

The area generated by moving the vector a from O along p is called the momental area of the Rotor a with regard to 0 as pole, the area being taken as a vector quantity.

The momental area is given therefore by [pa], and in the figure the aspect ort of the area is downwards through the plane of the paper.

The vector p may be any vector drawn from the pole O to the axis.

155. Properties of the Momental Area.

For a given Rotor the Momental Area will depend on the position of the pole and will in general change both in magnitude and aspect as the pole is moved.

If we are concerned with one plane only (as we shall be in this book) the aspect ort of the Momental Area is always perpendicular to the plane but may have a positive or negative sense.

Remembering the connection between the sense of the boundary and the aspect of the enclosed area (§§ 22 and 23) we say that a Momental Area [pa] is positive or negative according as the sense of the boundary is counter-clockwise or the reverse; the sense of the boundary being determined by p.

The farther the pole is from the rotor, the greater will be the magnitude of the Momental Area, and if the pole be on the rotor the Area will vanish.

We have then the following properties:

(i) The Momental Area may have any magnitude. (ii) It changes sign if the Pole crosses the axis of the Rotor.

(iii) It vanishes when the pole is on the rotor. From the properties of the Vector Product we have (iv) The Momental Area remains unaltered if the Rotor be shifted along its axis.

(v) It remains unaltered if the pole be shifted parallel to the Rotor.

or

If the Momental Area is zero, then either

156.

(a) the Pole is on the Rotor,

(b) the Rotor vanishes.

Theorem: If the momental area of a rotor vanishes for three poles not in a line, the rotor itself

must vanish.

We have just seen that if a momental area is zero for a pole P, then either P is on the rotor or the rotor is zero. If the momental area vanishes for a second pole Q. then the rotor must lie in PQ or it must be zero.

If it vanishes for a third pole not in PQ, then obviously the rotor itself must vanish.

157.

We now come to the combination of Rotors. Definition 1. Two systems of rotors are said to be equivalent if for every pole the sum of the momental areas of one system equals that of the other.

Definition 2. If one rotor can be found which is equivalent to a given system of rotors, it is called the resultant of the system.

Definition 3. If the sum of the momental areas of a system of rotors vanishes for every pole, the rotors are said to cancel or to be in equilibrium.

158. Theorem: If two rotors have equal momental areas for all poles, then the rotors themselves are equal.

Take any pole in the first rotor, then the momental area of that rotor vanishes and therefore that of the second must vanish. Hence the second rotor must pass through every point of the first, i.e. they must have the same axis. Having the same axis the vectors must be either equal or one must be a multiple of the other. From the equality of the momental areas the latter alternative is inadmissible.

Corollary. If a system of rotors has a resultant, it can have only one.

This follows at once from the preceding theorem and the definition of a resultant.

It is not proved that any system of rotors has a resultant, but if there is one, there can be only one.

159. Theorem: Two rotors whose axes meet have a resultant whose axis passes through their point of intersection and whose vector is the sum (or resultant) of the vectors of the given rotors.

В

B

α

Fig. 102.

Let C be the point of intersection of the axes, p the position vector of C from any pole O, a and B the vectors of the rotors.

Then since the Distributive Law holds for vector products

[pa] + [pB] = [p\a + ß].

Hence the sum of the momental areas of the rotors a and about any pole O is

equal to the momental area of the rotor whose axis passes through C and whose vector is a + B.

We can also say that the resultant of two intersecting rotors is found by the parallelogram law, viz. that it is given by the concurrent diagonal of the parallelogram of which the given rotors are adjacent sides.

Corollary 1. Any number of rotors through a common point have a resultant, which also passes through that point and whose vector is the sum (or resultant) of the vectors of the given rotors.

Corollary 2. If the vector-polygon for rotors through a common point be closed, the resultant vanishes and the rotors are in equilibrium.

160.

Note.

Note. In finding graphically the resultant of two rotors, the parallelogram should not be drawn. The sum of the two vectors is found in a separate figure.

This determines the Vector of the Resultant. The axis should then be drawn parallel to the vector through the intersection of the axes of the given rotors. The vector should not be set off on this axis.

For graphical work the following rules should always be adhered to.

A Rotor is graphically determined by drawing (i) its axis of indefinite length and (ii) its vector (not on the axis).

A system of Rotors is determined by drawing (i) the axes and (ii) the vectors, (in a separate figure) which are best added together at once to their Vector polygon.

Which Vector corresponds to each axis should be indicated by a suitable notation. For instance, if the vectors are denoted by a, B, y,... the corresponding axes may be denoted by a, b, c,.... In many cases a better notation can be employed as will be explained when required.

The whole process for finding the sum of a number of coplanar rotors consists in finding the sum of the vectors step by step and drawing the resultant at each step through the point of intersection of the axes of the rotors.