Through A draw any two rectangular axes Ax and Ay, let m, be any element of the lamina whose weight is w,, and let AM, and AN,, co-ordinates of m,, be represented by x, and y.. Then by equation (102), if a represent the angular velocity of the revolution of the body, the centrifugal force on the element m, is represented by w,Am,. Let now this

force, whose direction is Am, be whose directions are Ax and Ay.

g

resolved into two others, The former will be repre

a2

sented byw,Am,. cos. xAm,, or byw,,, and the latter

a2

a2

byw,Am, w, Am, cos. yAm,, or byw,y,; and the centrifugal

9

forces and all the other elements of the lamina being similarly resolved, we shall have obtained two sets of forces, those of the one set being parallel to Ax, and represented

Now if X and Y represent the resolved parts parallel to the directions of Ar and Ay, of the resultant of these two

if G, and G, represent the co-ordinates AG, and AG, of the centre of gravity G of the lamina, and W its weight (Art. 18.).

Now the whole centrifugal force F on the lamina is the resultant of these two sets of forces, and is represented by XY (Art. 11.),

where G is taken to represent the distance AG of the centre of gravity of the lamina from the axis of revolution.

Moreover, the direction of this resultant centrifugal force

is through A, since the direction of all its components are through that point

124. From the above formula, it is apparent that if a body revolving round a fixed axis be conceived to be divided into lamina by planes perpendicu lar to the axis, then the centrifugal force of each such laminæ is the same as it would have been if the whole of its weight had been collected in its centre of gravity; so that if the centres of gravity of all the lamina be in the same plane passing through the axis, then, since the centrifugal force on each lamina has its direction from the axis through the centre of gravity of that lamina, it follows that all the centrifugal forces of these laminæ are in the same plane, and that they are PARALLEL forces, so that their resultant is equal to their sum, those being taken with a negative sign which correspond to lamina whose centres of gravity are on the opposite side of the axis from the rest, and whose centrifugal forces are therefore in the opposite directions to those of the rest. Thus if F' represent the whole centrifugal force of such a mass, then F'WG. Now let W' represent the weight of the whole mass, and G' the distance of its centre of gravity from the axis, therefore ΣWG=W'G' ;

a2

In the case, then, of a revolving body capable of being divided into laminæ perpendicular to the axis of revolution, the centres of gravity of all of which lamina are in the same plane passing through the axis, the centrifugal force is the same as it would have been if the whole weight of the body had been collected in its centre of gravity, the same property obtaining in this case in respect to the whole body as obtains in respect to each of its individual lamina. Since, moreover, the centrifugal forces upon the lamina 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 (104) and (105) 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 lamina 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, WG=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 lamina 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 give. above.

THE PRINCIPLE OF VIRTUAL 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, Or, Oy, Oz; and if AC, AD, and AE, be the projections of AB upon these ares, 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, B, y, and the inclniations of AB to the same axes by a,, B, Y1,

..(Art. 12.) X=P cos. a, Y=P cos. B, Z=P cos. y; also AC =AB cos. a,, AD=AB cos. 6,, AE AB cos. Y1, .. X. AC-P. AB cos. a cos. a,, Y. AD=P. AB cos. B cos. B, Z.AE=P.AB cos. y cos. Y19

.X.AC+Y. AD+Z. AE=P. AB (cos, a cos. a,+cos. B cos. 6+cos. y cos. y1}.

But by a well-known theorem of trigonometry, cos. a cos. a1+cos. B cos. B1+cos. y cos. y,=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.*

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 appli cation 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

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, But the work of the sum of the works of X, Y, Z, must equal the work of P. X, as its point of. application moves through AB, is equal (Art. 52.) to the work of X through the projection of AB upon Ar, 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 A, 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.

directions in which the several forces applied to them act shall equal the aggregate of the work of those forces, this 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 words of these resolved forces respectively on any point, and Eu, the sum of all the works of the resolved forces of the set A, Zu, the sum of all the works of the forces of the set B, and Zu, of the set C. Now since the parallel forces of the set A have no 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 u,, which compose this sum, be taken with the positive or negative sign, according to the last mentioned condition.

Similarly, u,=0 and £u,=0, :. Σ(u,+u2+u ̧)=().

Now let U represent the actual work of that force P,, the works of whose components parallel to the three axes are represented by u,, u,, 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