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Digression on projection of areas.

If we imagine an area divided into any number of parts, the projections of these parts on any plane make up the projection of the whole. But in this statement it must be understood that the areas of partial projections are to be reckoned as positive if particular sides, which, for brevity, we may call the outside of the projected area and the front of the plane of projection, face the same way, and negative if they face oppositely.

Of course if the projected surface, or any part of it, be a plane area at right angles to the plane of projection, the projection vanishes. The projections of any two shells having a common edge, on any plane, are equal, but with the same, or opposite, signs as the case may be. Hence, by taking two such shells facing opposite ways, we see that the projection of a closed surface (or a shell with evanescent edge), on any plane, is nothing.

Equal areas in one plane, or in parallel planes, have equal projections on any plane, whatever may be their figures.

Hence the projection of any plane figure, or of any shell, edged by a plane figure, on another plane, is equal to its area, multiplied by the cosine of the angle at which its plane is inclined to the plane of projection. This angle is acute or obtuse, according as the outside of the projected area, and the front of plane of projection, face on the whole towards the same parts, or oppositely. Hence lines representing, as above described, moments about a point in different planes, are to be compounded as forces are.—See an analogous theorem in $ 96.

234. A Couple is a pair of equal forces acting in dissimilar directions in parallel lines. The Moment of a couple is the sum of the moments of its forces about any point in their plane, and is therefore equal to the product of either force into the shortest distance between their directions. This distance is called the Arm of the couple.

The Axis of a Couple is a line drawn from any chosen point of reference perpendicular to the plane of the couple, of such magnitude and in such direction as to represent the magnitude of the moment, and to indicate the direction in which the couple tends to turn. The most convenient rule for fulfilling the latter condition is this:-Hold a watch with its centre at the




point of reference, and with its plane parallel to the plane of Couplo. the couple. Then, according as the motion of the hands is contrary to or along with the direction in which the couple tends to turn, draw the axis of the couple through the face or through the back of the watch, from its centre. Thus a couple is completely represented by its axis; and couples are to be resolved and compounded by the same geometrical constructions performed with reference to their axes as forces or velocities, with reference to the lines directly representing them.

235. If we substitute, for the force in $ 232, a velocity, we Moment of have the moment of a velocity about a point; and by introducing the mass of the moving body as a factor, we have an important element of dynamical science, the Moment of Momen- Moment of tum. The laws of composition and resolution are the same as those already explained; but for the sake of some simple applications we give an elementary investigation. The moment of a rectilineal motion is the product of its Moment of

a rectilineal length into the distance of its line from the point.

displace. The moment of the resultant velocity of a particle about any point in the plane of the components is equal to the algebraic sum of the moments of the components, the proper sign of each moment being determined as above, $ 233. The same is of course true of moments of displacements, of moments of forces and of moments of momentum.

First, consider two component motions, AB and AC, and let for two AD be their resultant ($ 27). Their half moments round the motions, point ( are respectively the areas OAB, OCA. Now OCA, or motogether with half the area of the parallelogram CABD, is one plane. equal to OBD.

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Hence the sum of the two half moments their motogether with half the area of the parallelogram, is equal to proved AOB together with BOD, that is to say, to the area of the moment of whole figure OABD. But ABD, a part of this figure, is equal to half the area of

point in the parallelogram; and therefore the remainder, OAD, is equal to the sum of the two half moments. But OAD is half the moment of the resultant velocity round the point 0. Hence the moment of the Å





equal to the

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their resultant round any

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ber of moments in one

resultant is equal to the sum of the moments of the two components.

If there are any number of component rectilineal motions in

one plane, we may compound them in order, any two taken Any num. together first, then a third, and so on; and it follows that the

sum of their moments is equal to the moment of their resultant. plane com. It follows, of course, that the sum of the moments of


number pounded by addition. of component velocities, all in one plane, into which the velo

city of any point may be resolved, is equal to the moment of their resultant, round any point in their plane. It follows also, that if velocities, in different directions all in one plane, be successively given to a moving point, so that at any time its velocity is their resultant, the moment of its velocity at any time is the sum of the moments of all the velocities which have been successively given to it.

Cor.—If one of the components always passes through the point, its moment vanishes. This is the case of a motion in which the acceleration is directed to a fixed point, and we thus reproduce the theorem of $ 36, a, that in this case the areas described by the railius-vector are proportional to the times; for, as we have seen, the moment of velocity is double the area

traced out by the radius-vector in unit of time. Moment 236. The moment of the velocity of a point round any

axis round an axis. is the moment of the velocity of its projection on a plane per

pendicular to the axis, round the point in which the plane is cut by the axis.

The moment of the whole motion of a point during any

time, round any axis, is twice the area described in that time axis. by the radius-vector of its projection on a plane perpendicular to

that axis.

If we consider the conical area traced by the radius-vector drawn from any fixed point to a moving point whose motion is not confined to one plane, we see that the projection of this area on any plane through the fixed point is half of what we have just defined as the moment of the whole motion round an axis perpendicular to it through the fixed point. Of all these planes, there is one on which the projection of the area is greater

Moment of a whole inotion, round an


than on any other; and the projection of the conical area on Moment of any plane perpendicular to this plane, is equal to nothing, the motion, proper interpretation of positive and negative projections being nais, used.

If any number of moving points are given, we may similarly consider the conical surface described by the radius-vector of each drawn from one fixed point. The same statement applies to the projection of the many-sheeted conical surface, thus presented. The resultant axis of the whole motion in any finite Resultant time, round the fixed point of the motions of all the moving points, is a line through the fixed point perpendicular to the plane on which the area of the whole projection is greater than on any other plane; and the moment of the whole motion round the resultant axis, is twice the area of this projection.

The resultant axis and moment of velocity, of any number of moving points, relatively to any fixed point, are respectively the resultant axis of the whole motion during an infinitely short time, and its moment, divided by the time.

The moment of the whole motion round any axis, of the motion of any number of points during any time, is equal to the moment of the whole motion round the resultant axis through any point of the former axis, multiplied into the cosine of the angle between the two axes.

The resultant axis, relatively to any fixed point, of the whole motion of any number of moving points, and the moment of the whole motion round it, are deduced by the same elementary constructions from the resultant axes and moments of the individual points, or partial groups of points of the system, as the direction and magnitude of a resultant displacement are deduced from any given lines and magnitudes of component Moment of displacements.

Corresponding statements apply, of course, to the moments of velocity and of momentum.



237. If the point of application of a force be displaced Virtual through a small space, the resolved part of the displacement in the direction of the force has been called its Virtual Velocity.


Practical unit.

Virtual This is positive or negative according as the virtual velocity is velocity,

in the same, or in the opposite, direction to that of the force.

The product of the force, into the virtual velocity of its point of application, has been called the Virtual Moment of the force. These terms we have introduced since they stand in the history and developments of the science; but, as we shall show further on, they are inferior substitutes for a far more useful set of ideas clearly laid down by Newton.

238. A force is said to do work if its place of application has a positive component motion in its direction; and the work done by it is measured by the product of its amount into this component motion.

Thus, in lifting coals from a pit, the amount of work done is proportional to the weight of the coals lifted; that is, to the force overcome in raising them; and also to the height through which they are raised. The unit for the measurement of work adopted in practice by British engineers, is that required to overcome a force equal to the gravity of a pound through the space of a foot; and is called a Foot-Pound.

In purely scientific measurements, the unit of work is not the foot-pound, but the kinetic unit force ($ 225) acting through unit of space. Thus, for example, as we shall show further on, this unit is adopted in measuring the work done by an electric current, the units for electric and magnetic measurements being founded upon the kinetic unit force.

If the weight be raised obliquely, as, for instance, along a smooth inclined plane, the space through which the force has to be overcome is increased in the ratio of the length to the height of the plane; but the force to be overcome is not the whole gravity of the weight, but only the component of the gravity parallel to the plane; and this is less than the gravity

in the ratio of the height of the plane to its length. By Work of a multiplying these two expressions together, we find, as we force.

might expect, that the amount of work required is unchanged by the substitution of the oblique for the vertical path.

239. Generally, for any force, the work done during an infinitely small displacement of the point of application is the


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