. Mécanique céleste . quations [124]. * (57) These forces are similar to those used in [119 &c.], bringing the same [124a] terms from under the sign 2, as in llie preceding note. Now >S.(-r—)> ^•(l—) S.( -—j, [13], represent the force S, resolved in a direction parallel to x,y,z, and tlie composition of tliese three forces will again produce tlie single force S. Multiplying e11 these by 2 . m, it viiW follow that the three forces &.(-—; S .(—-\ .S. .m; S .i~\ ., in the directions parallel to x, y, z, will produce tire single force S . 2 . ?m, in the direction ofthe origin of tha
. Mécanique céleste . quations [124]. * (57) These forces are similar to those used in [119 &c.], bringing the same [124a] terms from under the sign 2, as in llie preceding note. Now >S.(-r—)> ^•(l—) S.( -—j, [13], represent the force S, resolved in a direction parallel to x,y,z, and tlie composition of tliese three forces will again produce tlie single force S. Multiplying e11 these by 2 . m, it viiW follow that the three forces &.(-—; S .(—-\ .S. .m; S .i~\ ., in the directions parallel to x, y, z, will produce tire single force S . 2 . ?m, in the direction ofthe origin of that force. f (57f«) Supjjose the body m to be placed at m,in tlie annexed figure, (which is die same as that inpage 13), upon the continuation of the line B b, sothat its rectangular co-ordinates may be CH=x,HB = y, B m = z, and let the co-ordinate Bmheintersected in b, by a plane CDb c passing throughthe centre of gravity of the system C, the ordinateB b being denoted by the accented letter z. Then. I. iii. §15.] CENTRE OF GRAVITY OF A SYSTEM OF BODIES. 87 and by multiplying tiiis distance by the mass of the body, the sum of theseproducts will be notJiing, in consequence of the equations (o) [124.] To determine the position of tJie cc iitrc; of gravity, let X, Y, Z, be its threeco-ordinates, referred to a given [loiiit ; .7-, y, z, the co-ordinates of m, referredto the same point ; x, t/, z, those of vt, and in llu; same manner for the rest,the equations (o) [124], will give* 0 = 2 . ?M . (.1- A) ; [ 125] but we have 2 . m . X^ X. s . m, x .m being the whole mass of the system ;hence we shall have In like manner, ^ [12fi] 2. my „ ?— i Z = — ? 2. m [127] the general equation of the plane [^I9c] z = A x ~\- B y, gives b m = {B m — n b = z — z) = z — A X — B if from tlie point m we let fall, upon the plane CDbc, a perpendicular p, thisperpendicular, or distance of the body m from the plane, will be equal to bm multiplied bythe sine
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