=

3

vector PA, PA + AA, and it passes through B1, hence it lies in B1B2. This we combine with the rotor in ɑ2 and get as resultant a rotor in B,B, with vector PA,, and so we continue till we get in BB a rotor which is the resultant of the rotor ẞ and all the given rotors, its vector being PA. We now cancel the rotor B by adding the rotor equal and opposite to it; viz. the rotor in b which has the vector -ẞ=AP. The final resultant therefore has the vector AA, AP+PA, and it passes through B. This proves the correctness of the construction.

We have now to see how far this construction is general.

It will be seen that the pole P of the vector-polygon, which may be anywhere, can always be so placed that no side of the link-polygon is parallel to the axis of the next rotor with which it is to be combined. (This is where the original construction failed.) To insure this P must not lie on one of the sides of the vector-polygon (or sides produced).

4

Similarly if P does not lie on the line AA, the first and last sides of the link-polygon will not be parallel, and will therefore meet in some point B, provided that the points A and A, are not coincident. If A and A, are coincident then PA and PA, would coincide, and the corresponding lines in the link-polygon would be parallel, in this case the vector-polygon is closed and the resultant vector vanishes. Hence,

A system of rotors in a plane has always a resultant if the vector-polygon is not closed; its vector is the sum of the vectors of the given rotors and its position is determined by the link-polygon.

165. If the vector-polygon be closed, then as before we place in the line b two rotors with vectors ß and – B. The first we combine with the rotors in a1, a, a, and a and get as their resultant a rotor in b' (parallel to b) having a vector

PA, B.

Hence the given system of rotors is equivalent to the

general, equivalent to a couple whose momental area can be determined by a link-polygon.

In this case the sum of the momental areas of the system of rotors is the same for all poles.

166. It may happen that the lines b and b' of the link-polygon coincide; in that case the momental area of the couple vanishes. Hence the sum of the momental areas of the given system of rotors is equal to zero for every pole, i.e. the rotors cancel or are in equilibrium.

The construction given in § 165 shews again that a couple can be replaced by any other couple in the same plane having the same momental area. For the pole P of the vector-polygon may be anywhere in the plane, hence PA=B may be any vector whatsoever in the plane, also the line b may have any position, so that one rotor of the resultant couple may be any rotor in the plane. But wherever b may be taken and whatever magnitude, direction and sense may have, the momental area [YB] is constant.

167. Parallel Rotors. Parallel rotors in a plane do not require any separate investigation, they constitute only a special case of the general theorem. Their vectors are like vectors and therefore are added by adding their magnitudes, the vector-polygon being a straight line in the direction of the rotors.

Hence parallel rotors have a resultant parallel to them unless the sum of their magnitudes vanishes, in which case there is either a resultant couple or equilibrium.

168. Our results may be summed up as follows.

Theorem. If a number of rotors are given in a plane, then three cases are possible.

(1) The vector-polygon is not closed: there is a resultant.

(2) The vector-polygon is closed: there is a resultant couple.

(3) The vector-polygon and the link-polygon are both closed the rotors are in equilibrium.

169. Since there is only one resultant for a given system of rotors, and since both the position of the first line in the link-polygon, and the position of the pole of the vector-polygon are arbitrary, we must obtain the same line BB1, if,

(a) keeping the pole fixed we draw the line b in any other position,

(b) we shift the pole.

We derive hence two important geometrical theorems. If with the same pole and vector-polygon we draw a series of link-polygons they will all

vector-polygon. Having drawn a link-polygon for one pole P, shift the pole to P' and proceed as follows.

Draw a line QQ, in the link-polygon parallel to PP' cutting BB in some convenient point Q, to the rotor PA in BB1 add the rotor P'P in QQ4, the resultant is P'A in QB', to this add the rotor AA, in a, and the rotor P'A, in BB is obtained and so on in the usual way.

Now instead of adding the rotor P'P to PA in BB, we might have added it to the rotor PA, in B,B. The same resultant P'A, in B,'B' must be found by either method and it follows therefore that  ̧ and  ̧Â1⁄2 must intersect at some point Q1 in QQ4.

2

Again, the rotor P'P might have been added to the rotor PA, in Â1⁄2 ̧, hence ¿Â1⁄2 and B1⁄2Ð ̧ must intersect at some point Q2 on QQ4.

2

3,

In this way we can see that all the corresponding sides of the two link-polygons intersect on QQ4.

If then we draw for the same set of rotors two different link-polygons by taking different poles for the vector-polygon, then the corresponding sides of the link-polygons intersect in points which lie on a straight line, parallel to the line joining the two poles.

These two link-polygons both determine the line of position of the resultant. We get then the following theorem.

If all the sides of a polygon turn about fixed points in a given straight line, while all the vertices but one move along fixed straight lines, then the remaining vertex will also describe a straight line.

EXERCISES X.

(1) A number of coplanar rotors should be taken at random, and the resultant or resultant couple found by the method of § 164. (2) Shew that the same result is obtained in (1) by

(a) altering the order of addition,

(b) by shifting the pole.

(3) Start with a closed vector-polygon and shew that there is a resultant couple.

(4) Alter (a) the first line of the link-polygon,

(b) the pole of the vector-polygon,

and shew that the Resultant Couple obtained in each case has the same momental area as in (3).

(5) Find the resultant of the parallel rotors 8 and 6 distant apart 3 inches.

(6) Find the resultant of the parallel rotors 8 and 6 distant apart 3 inches.

(7) Find the resultant couple of 4 rotors through A, B, C, D (equispaced) whose vector-polygon is a crossed quadrilateral.

(8) Find the resultant of the parallel rotors 2, 3, 4, 6, 1 distant apart 2, 5, 3 and 4 inches.

170. Moment of a Rotor. The momental area of a rotor with regard to any pole is sometimes called the Moment of the Rotor about that pole. The magnitude of the moment is given by the product of the magnitude of the rotor and the perpendicular from the pole on the

rotor.

171. Graphical Construction for the Measurement of Momental Areas.

Suppose we have only one rotor a and require to determine its momental area about some pole 0.

Let xx be the axis and a the magnitude of the rotor. Choose any pole P for the vector-polygon and join it to the end points A, A, of the vector a.