in respect to each of its individual laminæ. Since, moreover, the centrifugal forces upon the laminæ are parallel forces when their centres of gravity are all in the same plane passing through the axis of gravity, and since their directions are all in that plane, it follows (Art. 16.), that if we take any point O in the axis, and measure the moments of these parallel forces from that point, and call the perpendicular distance OA of any lamina BC from that point, and H the distance of their resultant from that point, then

The equations (93) and (94) determine the amount and the point of application of the resultant of the centrifugal forces upon the mass, upon the supposition that it can be divided into laminæ perpendicular to the axis of revolution, all of which have their centres of gravity in the same plane passing through the axis.

It is evident that this condition is satisfied, if the body be symmetrical as to a certain axis, and that axis be in the same plane with the axis of revolution, and therefore if it intersect or if it be parallel to the axis of revolution.

If, in the case we have supposed, ΣWG=O, that is, if the centre of gravity be in the axis of revolution, then the centrifugal force vanishes. This is evidently the case where a body revolves round its axis of symmetry.

125. If the centres of gravity of the lamina into which the body is divided by planes perpendicular to the axis of revolution be not in the same plane (as in the figure), then the centrifugal forces of the different laminæ will not lie in the same plane, but diverge from the axis in different directions round it. The amount and direction of their resultant cannot in this case be determined by the equations which have been given above.

THE PRINCIPLE OF VIRTUual Velocities.

126. If any pressure P, whose point of application A is made to move through the straight line AB, be resolved into three others X, Y, Z, in the directions of the three rectangular axes, Ox, Oy, Oz; and if AC, AD, and AE, be the projections of AB upon these axes, then the work of P through AB is equal to the sum of the works of X, Y, Z, through AC, AD, and AE respectively, or X. AC+Y. AD+Z. AE= P. AM.

Let the inclinations of the direction of P to the axes Ox, Oy, Oz respectively, be represented by a, ß, y, and the inclinations of AB to the same axes by a1, B1, 71,

.'. (Art. 12.) X=P cos. a, Y=P cos. ß, Z=P cos. y; also AC = AB cos. a1, AD=AB cos. 61, AE=AB cos. Y1,

=

.. X.AC=P.AB cos. a cos. a1, Y. AD=P. AB cos. ß cos. ß1, Z. AE=P. AB cos. y cos. Y1,

... X. AC+Y. AD+Z.AE=P. AB {cos. a cos. a1 +cos. ß cos. B1+cos. y cos. y}.

But by a well known theorem of trigonometry, cos. a cos. α, +cos. ß cos. B1+cos. y cos. Y1=cos. PAB,

... X. AC+Y. AD+Z.AE=P. AB cos. PAB; but AB cos. PAB=AM;

.. X. AC+Y. AD+Z. AE=P. AM. But (Art. 52.) the work of P through AM is equal to its work through AB. Therefore, &c.*

* This proposition may readily be deduced from Art. 53., for pressures equal and opposite to X, Y, Z, would just be in equilibrium with P, and these tending to move the point A in one direction along the line AB, P tends to move it in the opposite direction, therefore in the motion of the point A through AB, the sum of the works of X, Y, Z, must equal the work of P. But the work of X as its point of application moves through AB is equal (Art. 52.) to the work of X through the projection of AB

127. If any number of forces be in equilibrium (being in any way mechanically connected with one another), and if, subject to that connection, their different points of application be made to move, each through any exceedingly small distance, then the aggregate of the work of those forces, whose points of application are made to move towards the directions in which the several forces applied to them act, shall equal the aggregate of the work of those forces, the motions of whose points of application are opposed to the directions of the forces applied to them.

For let all the forces composing such a system be resolved into three sets of forces parallel to three rectangular axes, and let these three sets of parallel forces be represented by A, B, and C respectively. Then must the resultant of the parallel forces of each set equal nothing. For if any of these resultants had a finite value, then (by Art. 12.) the whole three sets of forces would have a resultant, which they cannot, since they are in equilibrium.

Now let the motion of the points of application of the forces be conceived so small that the amounts and directions of the forces may be made to vary, during the motion, only by an exceedingly small quantity, and so that the resolved forces upon any point of application may remain sensibly unchanged. Also let u,, u, u,, represent the works of these resolved forces respectively on any point, and Zu, the sum of all the works of the resolved forces of the set A, Σu, the sum of all the works of the forces of the set B, and Σu, of the set C. Now since the parallel forces of the set A have no

upon Ax, that is, through AC; similarly the work of Y, as its point of application moves through AB, is equal to its work through the projection of AB upon Ay, or through AD; and so of Z. The sum of the works of X, Y, and Z, as their point of application is made to move through AB, is therefore equal to what would have been the sum of their works had their points of application been made to move separately through AC, AD, AE; this last sum is therefore equal to the work of P through AB, which is equal to the work of P through AM, AM being the projection of AB upon the direction of P.

K

resultant, therefore (Art. 59.) the sum of the works of those forces of this set, whose points of application are moved towards the directions of their forces, is equal to the sum of the works of those whose points of application are moved from the directions of their forces, so that Eu,=0, if the values of u1, which compose this sum, be taken with the positive or negative sign, according to the last mentioned condition.

Similarly, Eu,=0 and Eu,=0, ..Σ(u1 + u2+uz)=0. Now let U represent the actual work of that force P1, the works of whose components parallel to the three axes are represented by u1, u2, u,; then by the last proposition,

in which expression U is to be taken positively or negatively according to the same condition as u,, u,, u,; that is, according as the work at each point is done in the direction of the corresponding force, or in a direction opposite to it. Hence therefore it follows, from the above equations, that the sum of the works in one of these directions equals their sum in the opposite direction. Therefore, &c.

The projection of the line described by the point of application of any force upon the direction of that force is called its VIRTUAL VELOCITY, so that the product of the force by its virtual velocity is in fact the work of that force; hence therefore, representing any force of the system by P, and its virtual velocity by p, we have Pp=U, and therefore, ΣPp=0, which is the principle of virtual velocities.*

128. If there be a system of forces such that their points of application being moved through certain consecutive positions, those forces are in all such positions in equilibrium, then in respect to any finite motion of the points of appli cation through that series of positions, the aggregate of the

This proof of the principle of virtual velocities is given here for the first time.

work of those forces, which act in the directions in which their several points of application are made to move, is equal to the aggregate of the work in the opposite direction.

This principle has been proved in the preceding proposi tion, only when the motions communicated to the several points of application are exceedingly small, so that the work done by each force is done only through an exceedingly small space. It extends, however, to the case in which each point of application is made to move, and the work of each force to be done, through any distance, however great, provided only that in all the different positions which the points of application of the forces of the system are thus made to take up, these forces be in equilibrium with one another; for it is evident that if there be a series of such positions immediately adjacent to one another, then the principle obtains in respect to each small motion from one of this set of positions into the adjacent one, and therefore in respect to the sum of all such small motions as may take place in the system in its passage from any one position into any other, that is, in respect to the whole motion of the system through the intervening series of positions. Therefore, &c.

THE PRINCIPLE OF VIS VIVA.

129. If the forces of any system be not in equilibrium with one another, then the difference between the aggregate work of those whose tendency is in the direction of the motions of their several points of application, and those whose tendency is in the opposite direction, is equal to one half the aggregate vis viva of the system.

In each of the consecutive positions which the bodies composing the system are made successively to take up, let there be applied to each body a force equal to the effective force (Art. 103.) upon that body, but in an opposite direction; every position will then become one of equilibrium.