are obtained for the motions of the system with regard to the planes xoz, yoz. These three equations, together with

are the general equations of the motions of a system of bodies which does not contain a fixed point.

159. When the bodies are independent of foreign forces, and only subject to their reciprocal attraction and to the force at o, the sum of the terms

m {Xy - Yx } + m' { X'y' — Y'x' },

arising from the mutual action of any two bodies in the system, m, m', is zero, by reason of the equality and opposition of action and reaction; and this is true for every such pair as m and m", m' and m'', &c. Iff be the distance of m from 0, F the force which urges the body m towards that origin, then

are its component forces; and when substituted in the preceding equations, F vanishes; the same may be shown with regard to m', m", &c. Hence the equations of areas are reduced to

Ση

yd x- xd y
dt2

}=

= 0,

As the first members of these equations are the sum of the masses of all the bodies of the system, respectively multiplied by the projec

tions of double the areas they describe on the co-ordinate planes, this sum is proportional to the time.

If the centre of gravity be the origin of the co-ordinates, the preceding equations may be expressed thus,

cdt=

c'dt =

c'dt=

Emm' {(xx) (dy' — dy) — (y' − y) (dx' — dx)}

Ση

Emm' {(2') (dx' dr) - (x' - x) (dz' - dz)}

1

Ση

Σmm' { (y' — y) (dz' — dz) — (z' — z) (dy' — dy) }
Ση

So that the principle of areas is reduced to depend on the co-ordinates of the mutual distances of the bodies of the system.

160. These results may be expressed by a diagram. Let m, m', m", fig. 44, &c., be a system of bodies revolving about o, the origin

in an indefinitely small time, represented by dt;

and let mon, m'on',

&c., be the projections of these areas on the plane roy. Then the equation

Em {xdy-ydx} = cdt,

shows that the sum of the products of twice the area mon by the mass m, twice the area m'on' by the mass m', twice m'on' by the mass m'', &c., is proportional to the element of the time in which they are described: whence it follows that the sum of the projections of the areas, each multiplied by the corresponding mass, is proportional to the finite time in which they are described. The other two equations express similar results for the areas projected on the planes

xoz, yoz.

161. The constancy of areas is evidently true for any plane whatever, since the position of the co-ordinate planes is arbitrary. The

three equations of areas give the space described by the bodies on each co-ordinate plane in value of the time: hence, if the time be known or assumed, the corresponding places of the bodies will be found on the three planes, and from thence their true positions in space may be determined, since that of the co-ordinate planes ist supposed to be known. It was shown, in article 132, that Σm { xdy — ydx},

Σm {zdx xdz},

are the pressures of the system, tending to make it turn round each of the axes of the co-ordinates: hence the principle of areas consists in this that the sum of the rotatory pressures which cause a system of bodies to revolve about a given point, is zero when the system is in equilibrio, and proportional to the time when the system is in motion.

162. Let us endeavour to ascertain whether any planes exist on which the sums of the areas are zero when the system is in motion. To solve this problem it is necessary to determine one set of coordinates in values of another.

If the sum of these quantities be taken, after multiplying the first by cos xox', the second by cos xoy', and the third by cos xoz', we shall have a' cos xox' + y' cos xoy' + z' cos xoz' =

x { cos xox' + cos xoy' + cos xoz' } = x.

Let oy, fig. 46, be the intersection of the old plane roy with the new r'oy'; and let be the inclination of these two planes; also let yox, yor' be represented by y and p. Values of the

hence, if

may be;

90° be put for p, the line or' will take the place of oy', the angle ror' will become roy', and the preceding equation will give cos xoy' cos 0 sin y cos - cos y sin p.

Cos roz' is found from the triangle whose three sides are the arcs intercepted by the angles yoz', yox, and roz'. The angle opposite to the last side is 90° — 0, yoz' = 90°

then the general equation becomes

If these expressions for the cosines be substituted in the value of

yox = 4,

cos roz' sin 0 sin y.

x, it becomes x = x' { cos 0 sin p sin

y' { cos 0 cos sin In the same manner, the values of y=x' { cos

y' { cos 0 cos y cos

2=

cos y sin

sin cos } +

+ sin y sin ø } + z' { sin 0 cos & }

x' { sin 0 sin } y' { sin 0 cos } + z' cos 0.

By substituting these values of x, y, z, in any equation, it will be transformed from the planes roy, roz, yoz, to the new planes x'oy', x'oz', y'oz'.

164. We have now the means of ascertaining whether, among the infinite number of co-ordinate planes whose origin is in o, the centre of gravity of a system of bodies, there be any on which the sums of the areas are zero. This may be known by substituting the preceding values of x, y, z, and their differentials in the equations of areas for the angles 0, 4, and being arbitrary, such values may be assumed for two of them as will make the sums of the projected areas on two of the co-ordinate planes zero; and if there be any

incongruity in this assumption, it will appear in the determination of the third angle, which in that case would involve some absurdity in the areas on the third plane. That, however, is by no means the case, for the sum of the areas on the third plane is then found to be a maximum. If the substitution be made, and the angles and so assumed that

Thus, in every system of revolving bodies, there

plane, on which the sum of the projected areas is and on every plane at right angles to it, they are zero. alone possesses that property.

does exist a

a maximum ;

One plane

165. If the attractive force at o were to cease, the bodies would move by the primitive impulse alone, and the principle of areas would be also true in this case; it even exists independently of any abrupt changes of motion or velocity, among the bodies; and also when the centre of gravity has a rectilinear motion in space. Indeed it follows as a matter of course, that all the properties which have been proved to exist in the motions of a system of bodies, whose centre of gravity is at rest, must equally exist, if that point has a uniform and rectilinear motion in space, since experience shows that the relative motions of a system of bodies, is independent of any motion common to them all.

Demonstration. However, that will readily appear, if x, y, z, be assumed, as the co-ordinates of o, the moveable centre of gravity estimated from a fixed