Therefore R-1 may, at any point of its direction be resolved into n pressures parallel and equal to P,, P., P„ . . . . . P„; if, therefore, n such pressures were applied to that point, they would just be held in equilibrium by a pressure equal and opposite to R-1. But R-1= 0; these n pressures would, therefore, be in equilibrium with one another if applied to this point.

Now it is evident, that if, being thus applied to this point, they would be in equilibrium, they would be in equilibrium if similarly applied to any other point. Therefore, &c.

THE POLYGON OF PRESSURES.

9. The conditions of the equilibrium of any number of pressures applied to a point.

Let OP,, OP,, OP,, &c., represent in mag nitude and direction pressures P1, P., &c., applied to the same point O. Complete the parallelogram OP, AP,, and draw its diagonal OA; then will OA represent in magnitude and direction the resultant of P, and Complete the parallelogram OABP,, then will OB represent in magnitude and direction the resultant of OA and P,; but OA is the resultant of P, and P2, therefore OB is the resultant of P., P, P.; similarly, if the parallelogram OBCP, be completed, its diagonal OC represents the resultant of OB and P., that is, of P1, P., P,, P., and in like manner OD, the diagonal of the parallelogram OCDP,, · represents the resultant of P,, P„, P„, P., P„.

Now let it be observed, that AP, is equal and parallel to OP, AB to OP,, BC to OP,, CD to OP, so that P,A, AB, BC, CD, represent P, P., P., P, respectively in magnitude, and are parallel to their directions. Moreover, OP, is in the direction of P, and represents it in magnitude, so that the sides OP,, PA, AB, BC, CD, of the polygon OP,, ABCDO represent the pressures P, P, P, P., P., respectively in magnitude, and are parallel to their directions; whilst the side OD, which completes that polygon, represents the resultant of those pressures in magnitude and direction.

If, therefore, the pressures P,, P,, P., P., P., be in equili brium, so that they have no resultant, then the side OD of the polygon must vanish, and the point D coincide with O. Thus, then, if any number of pressures be applied to a point

and lines be drawn parallel to the directions of those pres sures, and representing them in magnitude, so as to form sides of a polygon (care being taken to draw each line from the point where it unites with the preceding, towards the direction in which the corresponding pressure acts), then the line thus drawn parallel to the last pressure, and representing it in magnitude, will pass through the point from which the polygon commenced, and will just complete it if the pressures be in equilibrium; and if they be not in equilibrium, then this last line will not complete the polygon, and if a line be drawn completing it, that line will represent the resultant of all the pressures in magnitude and direction.

This principle is that of the POLYGON OF PRESSURES; it obtains in respect to pressures applied to the same point, whether they be in the same plane or not.

10. If any number of pressures in the same plane be in equilibrium, and each be resolved in directions parallel to any two rectangular axes, then the sum of all those resolved pressures, whose tendency is to communicate motion in one "direction along either axis, is equal to the sum of those whose tendency is in the opposite direction.

Let the polygon of pressures be formed in respect to any number of pressures, P1, P2, P, P,, in the same plane and in equilibrium (Arts. 8, 9), and let the sides of this polygon be projected on any straight line A in the same plane. Now it is evident, that the sum of the projections of those sides of the polygon which form that side of the figure which is nearest to Ac, is equal to the sum of the projections of those sides which form the opposite side of the polygon: moreover, that the former are those sides of the polygon which represent pressures tending to communicate motion from A towards e, or from left to right in respect to the line Az; and the latter, those which tend to communicate motion in the opposite direction. Now each projection is equal to the corresponding side of the polygon, multiplied by the cosine of its inclination to Ax. The sum of all those sides of the polygon which represent pressures tending to communicate motion from A towards a, multiplied each by the cosine of its inclination to Ax, is equal, therefore, to the sum of all the sides representing pressures whose tendency is in the opposite direction, each being similarly multiplied by the cosine of its inclination to Ax. Now the sides of the

polygon represent the pressures in magnitude, and are inclined at the same angles to Ax. Therefore, each pressure being multiplied by the cosine of its inclination to Aa, the sum of all these products, in respect to those which tend to communicate motion in one direction, equals the sum similarly taken in respect to those which tend to communicate motion in the opposite direction; or, if in taking this sum it be understood that each term into which there enters a pressure, whose tendency is from A towards a, is to be taken positively, whilst each into which there enters a pressure which tends from a towards A is to be taken negatively, then the sum of all these terms will equal zero; that is, calling the inclinations of the directions of P1, P„, P ̧ . . . P„ to Ax, α, α, α. . . . . an respectively,

P, cos. a,+P, cos.a, + P,cos.a, +.... + Pn cos. a2 =0...(5),

an

in which expression all those terms are to be taken negatively which include pressures, whose tendency is from x towards A.

This proposition being true in respect to any axis, Ax is' true in respect to another axis, to which the inclinations of the directions of the pressures are represented by B1, B2, B3, Br, so that,

P, cos. B, + P, cos. ẞ2+

1

....

+ P, cos. 3, =0.

Let this second axis be at right angles to the first:

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.. P, sin. a, + P, sin. a, + . . . . + Pn sin. an = 0.... (6); those terms in this equation, involving pressures which tend to communicate motion in one direction, in respect to the axis Ay being taken with the positive sign, and those which tend in the opposite direction with the negative sign.

If the pressures P1, P., &c. be each of them resolved into two others, one of which is parallel to the axis Ax, and the other to the axis Ay, it is evident that the pressures thus resolved parallel to As, will be represented by P, cos. a, P, cos. a, &c., and those resolved parallel to Ay, by P, sin. a, P, sin. a,, &c. Thus then it follows, that if any system of pressures in equilibrium be thus resolved parallel to two rectangular axes, the sum of those resolved pressures, whose tendency is in one direction along either

2

axis, is equal to the sum of those whose tendency is in the opposite direction.*

This condition, and that of the equality of moments, are necessary to the equilibrium of any number of pressures in the same plane, and they are together sufficient to that equi librium.

11. To determine the resultant of any number of pressures in the same plane.

....

If the pressures P, P, P1 be not in equilibrium, and have a resultant, then one side is wanting to complete the polygon of pressures, and that side represents the resultant of all the pressures in magnitude, and is parallel to its direction (Art. 9). Moreover it is evident, that in this case the sum of the projections on Ax (Art. 10) of those lines which form one side of the polygon, will be deficient of the sum of those of the lines which form the other side of the polygon, by the projection of this last deficient side; and therefore, that the sum of the resolved pressures acting in one direction along the line Ac, will be less than the sum of the resolved pressures in the opposite direction, by the resolved part of the resultant along this line. Now if R represent this resultant, and its inclination to Ax, then R cos. O is the resolved part of R in the direction of Az. Therefore the signs of the terms being understood as before, we have

R cos. 0=P, cos. a,+P, cos. ɑ2+

+Pn Cos. an (7).

And reasoning similarly in respect to the axis Ay, we have

Squaring these equations and adding them, and observing that R2 sin." 0+R' cos.' 0=R (sin.' 0+cos.'0) = R2, we have

R' (EP sin. a)+(EP cos. a)

(9),

where ΣP sin. a is taken to represent the sum P, sin. a1 + P, sin. a,+P, sin. a,+&c., and EP cos, a to represent the sum P, cos. a,+P, cos. a,+P, cos. a2+ &c.

Dividing equation (8) by equation (7),

Thus then by equation (9) the magnitude of the resultant

*Note (f) Ed. App.

R is known, and by equation (10) its inclination to the axig Ax is known. In order completely to determine it, we have yet to find the perpendicular distance at which it acts from the given point A. For this we must have recourse to the condition of the equality of moments (Art. 7).

2

If the sum of the moments of those of the pressures, P1, P1. . . . Pn, which tend to turn the system in one direction about A, do not equal the sum of the moments of those which tend to turn it the other way, then a pressure being applied to the system, equal and opposite to the resultant R, will bring about the equality of these two sums, so that the moment of R must be equal to the difference of these sums. Let then p equal the perpendicular distance of the direction of R from A. Therefore

Rp=mt P,+mt P2+m' P,+ . . . . +mt Pr... (11), in the second member of which equation the moments of those pressures are to be taken negatively, which tend to communicate motion round A towards the left.

Dividing both sides by R we have

Thus then by equations (9), (10), (12), the magnitude of the resultant R, its inclination to the given axis Ax, and the perpendicular distance of its direction from the point A, are known; and thus the resultant pressure is completely determined in magnitude and direction.

THE PARALLELOPIPEDON OF PRESSURES.

12. Three pressures, P1, P2, P3, being applied to the sume point A, in directions xA, yA, zA, which are not in the same plane, it is required to determine their resultant. Take the lines P, A, P, A, P, A, to represent the pressures P., P., P., in magnitude and direction.

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2

Complete the parallelopipedon RP,P,P„ of which AP,, AP,, AP, are adjacent edges, and draw its diagonal RA; then will RA represent the resultant of P1, P2, P1, in direction and magnitude. For since PSP,A is a parallelogram, whose adjacent sides P, A, P, A, represent the presures P, and P, in magnitude and direction, therefore its diagonal