53. If any number of pressures P1, P., P., be applied to the same point A, and remain constant and parallel to themselves, whilst the point A is made to move through the straight line AB, then the whole work done is equal to the sum of the works of the different pressures resolved in the directions of those pressures, each being taken negatively whose point of application is made to move in an opposite direction to the pressure upon it.

α α

1

2

Let a,, a, a,, &c. represent the inclinations of the pressures P1, P,, &c. to the line AB, then will the resolved parts of these pressures in the direction of that line be P, cos. a1, P, cos. a, P, cos. a,, &c. and they will be equivalent to a single pressure in the direction of that line represented by P, cos. a,+P, cos. a2+P, cos. ",, &c. in which sum all those terms are to be taken negatively which involve pres sures whose direction is from B towards A (since the single pressure from A towards B is manifestly equal to the difference between the sum of those resolved pressures which act in that direction, and those in the opposite direction). Therefore the whole work is equal to P, cos. a, + P, cos. a, + P, }. AB=P,. AB cos. α, + P, . AB cos. ", +P,AB cos. a2+ . . = P, . BM, +P ̧ . B„M‍+ P ̧ . BM,+ .; in which expression the successive terms are the works of the different pressures resolved in the several directions of those pressures, each being taken positively or negatively, according as the direction of the corresponding pressure is towards the direction of the motion or opposite to it.

3

Thus if U represent the whole work and U, and U, the sums of those done in opposite directions, then

54. If any number of pressures applied to a point be in equilibrium, and their point of application be moved, the whole work done by these pressures in the direction of the motion will equal the whole work done in the opposite direction.

For if the pressures P1, P2, P., &c. (Art. 53) be in equilibrium, then the sums of their resolved pressures in opposite *Note (n) Ed. App.

directions along AB will be equal (Art. 10); therefore the whole work U along AB, which by the last proposition is equal to the work of a pressure represented by the difference of these sums, will equal nothing, therefore 0U,- U„ therefore U,U,, that is, the whole work done in one direction along AB, by the pressures P1, P., &c. is equal to the whole work done in the opposite direction.

1

55. If a body be acted upon by a force whose direction is always towards a certain point S, called a centre of force, and be made to describe any given curve PA in a direction opposed to the action of that force, and Sp be measured on SA equal to SP, then will the work done in moving the body through the curve PA be equal to that which would be necessary to move it in a straight line from p to A.

For suppose the curve PA to be a portion of a polygon of an infinite number of sides, PP,, P,P,, &c. Through the points P, P., P., &c. describe circular ares with the radii SP, SP,, SP, &c. and let them intersect SA in P, P, P., &c. Then since PP, is exceedingly small, the force may be considered to act throughout this space always in a direction parallel to SP,; therefore the work done through PP, is equal to the work which must be done to move the body through the distance mP, (Art: 52.), since mP, is the projection of PP, upon the direction SP, of the force. But mP,=pp; therefore the work done through PP, is equal to that which would be required to move the body along the line SA through the distance pp,; and similarly the work done through P,P, is equal to that which must be done to move the body through PP, so that the work through PP, is equal to that through pp, and so of all other points in the curve. Therefore the work through PA is equal to that through pA.* Therefore, &c. [Q.E.D.]

Of course it is in this proposition supposed that the force, if it be not constant, is dependant for its amount only on the distance of the point at which it acts from the centre of force S; so that the distances of p and P from S being the same, the force at p is equal to that at P; similarly the force at p is equal to that at P1, the force at p. equal to that at P2, &c.

56. If She at an exceedingly great distance as compared with AP, then all the lines drawn from S to AP may be considered parallel. This is the case with the force of gravity at the surface of the earth, which tends towards a point, the earth's centre, situated at an exceedingly great distance, as compared with any of the distances through which the work of mechanical agents is usually estimated.

Thus then it follows that the work necessary to move a heavy body up any curve PA, or inclined plane, is the same as would be necessary to raise it in a vertical line pA to the same height.

The dimensions of the body are here supposed to be exceeding small. If it be of considerable dimensions, then whatever be the height through which its centre of gravity is raised along the curve, the work expended is the same (Art 60.) as though the centre of gravity were raised vertically to that height.*

57. In the preceding propositions the work has been estimated on the supposition that the body is made to move so as to increase its distance from the centre S, or in a direction opposed to that of the force impelling it towards S. It is evident, nevertheless that the work would have been precisely the same, if instead of the body moving from P to A it had moved from A to P, provided only that in this last case there were applied to it at every point such a force as would prevent its motion from being accelerated by the force continually impelling it towards S; for it is evident that to prevent this acceleration, there must continually be applied to the body a force in a direction from S equal to that by which it is attracted towards it; and the work of such a force is manifestly the same, provided the path be the same, whether the body move in one direction or the other along that path, being in the two cases the work of the same force over the same space, but in opposite directions.

* The only force acting upon the body is in this proposition supposed to be that acting towards S. No account is taken of friction or any other forces which oppose themselves to its motion.

58. If there be any number of parallel pressures, P1, P2, P3, &c. whose points of application are transferred, each through any given distance from one position to another, then is the work which would be necessary to transfer their resultant through a space equal to that by which their centre of pressure is displaced in this change of position, equal to the difference between the aggregate work of those pressures whose points of application have been moved in the directions in which the pressures applied to them act, and those whose points of application have been moved in the opposite directions to their pressures.

For (Art. 17.), if y1, Y., Y., &c. represent the distances of the points of application of these pressures from any given plane in their first position, and h the distance of their centre of pressure from that plane, and if Y,, Y, Y, &c. and H represent the corresponding distances in the second position, and if P,, P,, P., &c. be taken positively or negatively according as their directions are from or towards the given plane, h P,+P2+P2+ . . . } =P1y1+P2y2+P2y2

and HP,+P,+P,+ .... }=P,Y,+P,Y,+P,Y,+

.. (II—h) {P,+P2+P,+ . } = P, (Y,—y1)+P2 (ˇ,—Y2) +P ̧ (Y,−ys) +

3

(42);

in the second member of which equation the several terms are evidently positive or negative, according as the pressure P corresponding to each, and the difference Y-y of its dis tances from the plane in its two positions, have the same or contrary signs. Now by supposition Pis positive or negative according as it acts from or towards the plane; also Y-y is evidently positive or negative according as the point of application of P is moved from or towards the plane; each term is therefore positive or negative, according as the corresponding point of application is transferred in a direction towards that in which its applied pressure acts, or in the opposite direction.

Let

Now the plane from which the distances of the points of application are measured may be any plane whatever. it be a plane perpendicular to the directions of the pressures.

Let Axy represent this plane, and let P P' represent the two positions of the point of application of the pressure P (the path described by it between these two positions having been any whatever). Let MP and M'P' represent the perpendicular distances of the points P and P' from the plane, and draw Pm from P perpendicular to M'P'. Then P(Y-y)=P(M'P'-MP)=P. mP'; but by Art. 55., P. mP' equals the work of P as its point of application is transferred from P to P'. Thus each term of the second member of equation (42) represents the work of the corresponding pressure, so that if Lu,, represent the aggregate work of those pressures whose points of application are transferred towards the directions in which the pressures act, and Eu, the work of those whose points of application are moved opposite to the directions in which they severally act, then the second member of the equation is represented by Eu, — Zu. Moreover the first member of the equation is evidently the work necessary to transfer the resultant pressure P1+ P+P, &c. through the distance H-h, which is that by which the centre of pressure is removed from or towards the given plane, so that if U represent the quantity of work necessary to make this transfer of the centre of pressure,

=

... =0.

59. If the sum of those parallel pressures whose tendency is in one direction equal the sum of those whose tendency is in the opposite direction, then P,+P,+P ̧+ In this case, therefore, U=0, therefore Eu,-Eu,=0, therefore Lu, Lu,; so that when in any system of parallel pressures the sum of those whose tendency is in one direction equals the sum of those whose tendency is in the opposite direc tion, then the aggregate work of those whose points of appli cation are moved in the directions of the pressure severally applied to them is equal to the aggregate work of those whose points of application are moved in the opposite directions.

This case manifestly obtains when the parallel pressures are in EQUILIBRIUM, the sum of those whose tendency is in one direction then equalling the sum of those whose tendency is in the opposite direction, since otherwise, when applied to a point, these pressures could not be in equilibrium about that point (Art. 8.).