that is (by considering the masses m, n, o, as representing the forces, thevelocity u being common), the distance of the centre of gravity of several bodies from an assumed straight line, is found by dividing the sum of the moments of these bodies (taken with respect to this line) by the sum of the masses.

Let us now conceive the system of bodies m, n, o, reversed in such a manner that FA", instead of being horizontal, shall become vertical, &c.; it is apparent, that in order to find the distance of the resultant from the line FA", now vertical, it will be necessary to take the sum of the moments with respect to FA", and to divide this sum by the sum of the masses; which gives

Having found the distance of the pointG from two fixed known lines Top. 1. FA', FC', the position of this centre G is evidently determined.

It is here taken for granted that the distances A'A, A"A, B'B, BB", &c., are known, since the point through which FA", FC', are drawn, is assumed at pleasure.

77. If the distances A'A, B'B, &c., are each zero; that is, if all the bodies are in the same straight line FA," the sum of the moments with respect to this line is zero; the distance GG is therefore zero. Accordingly, if several bodies, considered as points, are in the same straight line, their common centre of gravity is also in this line.

78. If the lines FA", FC', are either of them drawn in such a manner as to have bodies situated on each side of it, instead of the sum of the moments, we should say the sum of the moments that are found on one side, minus the sum of the moments that are found on the other side. As to the denominator of the fraction which expresses the distance of the centre of gravity, it will always be composed of the sum of the masses, since all the forces, by the nature of gravity, act in the same direction. What is here said is applicable to any number of bodies, which, being considered as points, are situated in the same plane.

The lines FA", FC', are called the axes of the moments.

79. If now we suppose the point F, which we at first took arbitrarily, to be in G, G'G and G'G become each equal to zero. Therefore the sum of the moments with respect to FA", and the sum of the moments with respect to FC', must in this case be each equal to zero..

80. We now say, that if the sum of the moments of several bodies with respect to the straight line TS, passing through the Fig. 28. point G, is equal to zero; and the sum of the moments with respect to the straight line DE, perpendicular to TS, and passing also through G, is in like manner equal to zero; the sum of the moments with respect to any other straight line LH, passing through the same point G, will also be equal to zero.

Indeed, having let fall upon the lines DE, TS, LH, the perpendiculars AA', AA", AA"; if we suppose that the point I is that in which AA meets LH, from the right-angled triangle GA'I, we have

Now from the right-angled triangle IAA", we have, radius being supposed equal to 1,

1: AI :: sin AIA": AA""

:: cos DGL: AA" = AI x cos DGL;

that is, substituting for AI its value above found,

AA" = AA' cos DGL — AA' × sin DGL;

hence, if we multiply by the mass m to obtain the moment, we shall have

m × AA" = m x AA' x cos DGL

mx AA" × sin DGL;

in other words, the moment of the body m with respect to the axis LH, is equal to the cosine of the angle DGL, multiplied by the sum of the moments with respect to the axis DE, minus the

68.

sine of the same angle DGL, multiplied by the sum of the moments with respect to the axis TS.

Now it is manifest, that with regard to any other body n, we should arrive at a similar result, with the exception only of the signs according to which the bodies are on the same or on different sides of LH. Consequently, if we take the sum of all the moments with respect to the axis LH, we shall find that it is equal to the cosine of the angle DGL, multiplied by the sum of the moments with respect to DE, minus the sine of the angle DGL, multiplied by the sum of the moments with respect to TS. But each of these two last sums is by supposition equal to zero; consequently their products by the cosine and sine respectively of the angle DGL, will be each equal to zero; therefore, also, the sum of the moments with respect to any axis whatever LH, which passes through the centre of gravity G, is equal to zero.

81. Hence we infer that the resultant action of all the particular actions of gravity, which are exerted upon the several parts of a system of bodies, passes always through the same point of this system, whatever be its position; for it is not with respect to the direction of the resultant that the sum of the moments of the several parallel forces may be equal to zero.

Moreover, although the inquiry hitherto has been only respecting bodies whose centres of gravity are in the same plane, the method is not the less applicable to the case where the parts of the system are in different planes.

82. If the bodies, still regarded as points, are not in the same Fig. 23. plane, let us imagine a horizontal plane XZ, and from each of the gravitating points p, q, r, let the vertical lines Ap, Bq, Cr, be supposed to be drawn; and in order to determine the point E, through which passes the resultant o E, in the direction of which must be the centre of gravity, we take the moments with respect to two fixed lines FX, FZ, assumed in the horizontal. plane, perpendicular to each other; we take, I say, the sum of the moments, as if the bodies were all in this horizontal plane; and having divided each of the two sums of moments by the sum of the masses or forces p, q, r, we shall have the two distances EE, EE. It will only remain, therefore, to find at what distance EG, below the horizontal plane, this centre is situated.

Now if we imagine the figure reversed, the plane XZ becoming vertical, and ZV horizontal, it will be seen that in order to determine the distance E'G', corresponding and equal to EG, the distance sought, it is necessary, according to the method above pursued, to take the sum of the moments with respect to ZF, as if the bodies were all in the plane ZV, and to divide this sum by the sum of the masses; we have then every thing that is requisite in order to fix the position of the centre of gravity.

83. Hence, by recapitulating what we have said, this prob lem reduces itself to the following particulars;

(1.) When the several bodies, considered as points, are situated in the same straight line, we take the sum of the moments with Fig. 29. respect to a fixed point F, assumed arbitrarily in this line, and divide this sum by the sum of the masses, and the quotient will be the distance of the centre of gravity G from the point F.

(2.) When the several bodies, considered as points, are all in the same plane; through a point F, taken arbitrarily in this Fig. 27, plane, we suppose two lines FA", FC', to be drawn at right angles to each other; and having let fall perpendiculars upon each of these two lines from each gravitating point, we imagine that these gravitating points are applied successively to the lines FA", FC, where their perpendiculars respectively fall. We then seek, as in the case just stated, what would be the centre of gravity G" in FA", and what would be the centre of gravity Gin FC; drawing lastly through these two points the lines G"G, G'G, parallel respectively to FC, FA", and their point of meeting G will be the centre of gravity sought.

(3.) When the several bodies, considered as points, are in different planes, we imagine three planes, one horizontal, and Fig. 23. the two others vertical and perpendicular to each other. From each gravitating point we suppose perpendiculars let fall upon each of these three planes; we then take the sum of the moments with respect to each plane, and dividing each of these sums by the sum of the masses, we shall have the three distances of the centre of gravity from the three planes respectively.

84. It must be recollected, moreover, in what is above said, that when the bodies are on different sides of the line or plane with respect to which the moments are considered, it is necessaMech.

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ry to take with contrary signs the moments of bodies that are found on opposite sides.

85. We will here make a remark, that is suggested by what has been said, and which will enable us to abridge, in many cases, the process of finding the centre of gravity, as well as the solution of other problems.

Since the distance of the centre of gravity is expressed by the sum of the moments divided by the sum of the masses, if this centre happen to be in the point, line, or plane, with respect to which the moments are considered, the distance being zero, the sum of the moments must also be zero. Therefore, the sum of the moments with respect to any such plane as may pass through the centre of gravity is zero.

86. Hitherto we have considered bodies as so many points, and we have seen how the centre of gravity of all these points may be determined, whatever be their number and position. Now a body of any size or figure whatever, being only an assemblage of other bodies or material parts, which may be considered as points, it follows that, by the method above pursued, we may determine the centre of gravity of a body of any figure whatever.

Also, since the centre of gravity is simply the point through which passes the resultant of all the particular efforts made by the several parts of a body in virtue of their gravity, and since this resultant is equal to the sum of all these particular efforts; it follows, that we may in all cases suppose the whole weight of a body united at its centre of gravity, and the weight would have the same effect upon this point, when thus united, that it would have in its actual state of distribution through all parts of the body.

87. When, therefore, it is proposed to find the common centre of gravity of several masses of whatever figure, we begin by seeking the centre of gravity of each of these masses, which is attended with no difficulty. Then, the weight of these masses being considered as united each at its centre of gravity, we seek the common centre of gravity, as if all these bodies were points situated where each has its particular centre of gravity.