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those which tend in the opposite direction are to be taken negatively.

The line RN, represents the perpendicular distance from the given plane of a point through which the resultant of all the pressures P1, P, . . . . P, passes. In the same manner may be determined the distance of this point from any other plane. Let this distance be thus determined in respect to three given planes at right angles to one another. Its actual position in space will then be known. Thus then we shall know a point through which the resultant of all the pressures passes, also the direction of that resultant, for it is parallel to the common direction of all the pressures, and we shall know its amount, for it is equal to the sum of all the pressures with their proper signs. Thus then the resultant pressure will be completely known. The point R, is called the CENTRE OF PARALLEL PRESSURES.

18. The product of any pressure by its perpendicular distance from a plane (or rather the product of the number of units in the pressure by the number of units in the perpendicular), is called the moment of the pressure, in respect to that plane. Whence it follows from equation (17) that the sum of the moments of any number of parallel pressures in respect to a given plane is equal to the moment of their resultant in respect to that plane.

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19. It is evident, from equation (17), that the distance R-1 N of the centre of pressure of any number of parallel pressures from a given plane, is independent of the directions of these parallel pressures, and is dependent wholly upon their amounts and the perpendicular distances P,M,, PM2, &c. of their points of application from the given plane.

So that if the directions of the pressures were changed, provided that their amounts and points of application remained the same, their centre of pressure, determined as above, would remain unchanged; that is, the resultant,

although it would alter its direction with the directions of the component pressures, would, nevertheless, always pass through the same point.

The weights of any number of different bodies or different. parts of the same body, constitute a system of parallel pressures; the direction, therefore, through this system of the resultant weight may be determined by the preceding proposition; their centre of pressure is their centre of gravity.

THE CENTRE OF GRAVITY.

20. The resultant of the weights of any number of bodies or parts of the same body united into a system of invariable form passes through the same point in it, into whatever position it may be turned.

For the effect of turning it into different positions is to cause the directions of the weights of its parts to traverse the heavy body or system in different directions, at one time lengthwise for instance, at another across, at another obliquely; and the effect upon the direction of the resultant weight through the body, produced by thus turning it into different positions, and thereby changing the directions in which the weights of its component parts traverse its mass, is manifestly the same as would be produced, if without altering the position of the body, the direction of gravity could be changed so as, for instance, to make it at one time traverse that body longitudinally, at another obliquely, at a third transversely. But by Article 19. this last mentioned change, altering the common direction of the parallel pressures through the body without altering their amounts or their points of application, would not alter the position of their centre of pressure in the body; therefore, neither would the first mentioned change. Whence it follows that the centre of pressure of the weights of the parts of a heavy body, or of a system of invariable form, does not alter its position in the body, whatever may be the position into which

the body is turned; or in other words, that the resultant of the weights of its parts passes always through the same point in the body or system in whatever position it may be placed.

This point, through which the resultant of the weights of the parts of a body, or system of bodies of invariable form, passes, in whatever position it is placed; or, if it be a body or system of variable form, through which the resultant would pass, in whatever position it were placed, if it became rigid or invariable in its form, is called the CENTRE OF GRAVITY.

21. Since the weights of the parts of a body act in parallel directions, and all tend in the same direction, therefore their resultant is equal to their sum. Now, the resultant of the weights of the parts of the body would produce, singly, the same effect as it regards the conditions of the equilibrium of the body, that the weights of its parts actually do collectively, and this weight is equal to the sum of the weights of the parts, that is, to the whole weight of the body, and in every position it acts vertically downwards through the same point in the body, viz. the centre of gravity. Thus then it follows, that in every position of the body and under every circumstance, the weights of its parts produce the same effect in respect to the conditions of its equilibrium, as though they were all collected in and acted through that one point of it· its centre of gravity.*

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*That the resultant of the weights of all the parts of a rigid body passes in all the positions of that body through the same point in it is a property of many and most important uses in the mechanism of the universe, as well as in the practice of the arts; another proof of it is therefore subjoined, which may be more satisfactory to some readers than that

given in the text. The system being rigid, the distance P1, P2, of the points of application of any two of the pressures remains the same, into whatever position the body may be turned: the only difference produced in the circumstance under which they are applied is an alteration in the inclinations of these pressures to the

line P1, P2: now being weights, the directions of these pressures always remain parallel to one another, whatever may be their inclination; thus

22. To determine the position of the centre of gravity of two weights, P, and P2, forming part of a rigid system.

Let it be represented by G. Then since the resultant of P1 and P, passes through G, we have by equation

1

2

(16.), taking P, as the point from which the mo

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23. It is required to determine the centre of gravity of three weights P1, P2, P3, not in the same straight line, and forming part of a rigid system.

Find the centre of gravity G1, of P, and P2, as in the last proposition. Suppose the weights P, and P to be collected in G1, and find as before the common centre of gravity G of this weight P1 + P2, so

1

collected in G1, and the third weight P3. It is

evident that this point G, is the centre of gravity required.

then it follows by the principle of the equality of moments (Art. 15.), that P1+P ̧.·P,R=P,. P,P2, so that for every such inclination of the pressures to P, P, the line PR, is of the same length, and the point R, therefore the same point; therefore, the line PR, is always the same line in the body; and R, which equals P1+P, is always the same pressure, as also is Ps, and these pressures always remain parallel, therefore, for the same reason as before, R, is always the same point in the body in whatever position it may be turned, and so of R3, R. . . . and R. That is, in every position of the body, the resultant of the weights of its parts passes through the same point R, in it. Since the resultant of the weights of the parts of a body always passes through its centre of gravity, it is evident, that a single force applied at that point equal and opposite to this resultant, that is, equal in amount to the whole weight of the body, and in a direction vertically upwards, would in every position of the body sustain it. This property of the centre of gravity, viz. that it is a point in the body where a single force would support it, is sometimes taken as the definition of it.

3

Since G, is the centre of gravity of P, and P,+P, collected in G1, we have by the last proposition

G1G2. P1+P2+P ̧=G1P ̧ . P ̧,

2

1

2

3

G,P3. P3

3

.:. G,G,=P1+P2+P ̧

1

2

3

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24. To find the centre of gravity of four weights not in the same plane.

Let P1, P2, P3, P4, represent these weights; find the centre of gravity G of the weights P1, P2, P3, as in the last proposition; suppose these three weights to be collected in G2, and then find the centre of gravity G, of the weight thus collected in G, and P. G, will be the centre of gravity required, and since G, is the centre of gravity of P1 acting at the point P1, and of P1+ P2+P3 collected at G2,

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If all these weights be equal, then by the above equation,

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25. THE CENTRE OF GRAVITY OF A TRiangle.

Let the sides AB and BC of the triangular lamina ABC be bisected in E and D, and the lines CE and AD drawn to the opposite angles, then is the intersection G of these lines the centre of gravity of the triangle: for the triangle may be supposed to be made up of exceedingly narrow rectangular strips or bands,

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