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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 is
supposed to be known. It was shown, in article 132, that
Em { xdy – ydr},

Em zdxxdz},
Σm {zdy - ydz },

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


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.

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xx cos xox'.

ox ox :: 1: cos xox'
ox oy' :: 1: cos xoy'
ot: oz ::1: cos roz

or y' x cos xoy'.
or 2 = a cos 202,

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

x { cos2 xox' + cos2 xoy' + cos2 xoz' } = x.

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

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fig. 46.

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cos xox = cos 0 sin

sin + cos y cos .

This equation exists, whatever the values of and may be; hence, if +90° be put for p, the line or' will take the place of oy', the angle xox' will become roy', and the preceding equation will give cos xoy' = cos 0 sin 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, 7oz' = 90° then the general equation becomes

yox = 4,


cos roz' sin ◊ sin ¥.


+ cos y cos } +

If these expressions for the cosines be substituted in the value of x, it becomes = x' { cos 0 sin sin y' { cos e cos sin ¥ cos y sin } + z' sin 0 sin y. In the same manner, the values of y and z are found to be y=x' { cos o cos sin sin cos } +

} + z' { sin cos y }


y' { cos cos y cos + sin y sin x' { sin 0 sin } y' { sin cos } + z' cos 0. By substituting these values of x, y, z, in any equation, it will be transformed from the planes roy, xoz, 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 6, 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

cosines of xox', xoy', xoz', must
be found in terms of 0, 4, and
In the right-angled tri-
angle yxx', the sides yr, yx',
are and, and the angle
opposite the side rx is 0: —
hence, by spherical trigonome-

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 y and ℗ so assumed that c'



sino cos

√ c2 + c22 + c/2'

√ c2 + c22 + c'2'

cos 0=

it follows that



√ c2 + c22 + c2

Em x'dy' - y'dr'

dt z'dr


= √ c2 + c22 + c!!2,

Σm y'dz' — x'dy'




= 0,


Thus, in every system of revolving bodies, there does exist a plane, on which the sum of the projected areas is a maximum ; and on every plane at right angles to it, they are zero. One plane alone possesses that property.

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.

= 0.

fig. 47.

Demonstration.-However, that will readily appear, if 7, y, z, be assumed, as the co-ordinates of o, the moveable centre of gravity estimated from a fixed point P, fig. 47, and if oA, AB, Bm, or x', y', z', be the co-ordinates of m, one of the bodies of the system with regard to the moveable point o. Then the co-ordinates of m relatively

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to P will be + x', ÿ + y', ≈ + z'. If these be put instead of x, y, z, in the different equations relative to the motions of a system, by attending to the properties of the centre of gravity, x, y, z, vanish from these equations, which then become independent of them. If ï + x', ÿ + y', z + z' be put for x, y, z, in equations (25), they become

Σ.m { di + dx' }

Em Xdt


Σ.m { d'y + d'y' }

Em Ydt2


E.m{ dz + d2z' }

Σ.m Zdt = 0.

But when the centre of gravity has a rectilinear and uniform motion in space, it has been shown, that d27 dt

dex dt2

= 0;

= 0;

= 0;

which reduces the preceding equations to their original form, namely, dex' day' dt dt2


= ΣmX, Σm.

= ΣmY, Σm.

= ΣmZ.

If the same substitution be made in

it becomes'y'


Σm {xd'y = yd'x} = Em (xY




In the same manner it may be shown that

ɛ.m. {z'd2x'




+ Em, x'd3y' — y'd2x'



= Em. (Yx' - Xy') + a.ΣmY-y. EmX.

But in consequence of the preceding equations it is reduced to

Σ.m. {x'd'y' = y'd'x'}

=Σ.m.(x'Y - y'X.


{z'd2x' + x'd°z' } = Σ.m. (z'X – x'Z),


ɛ.m. {y'd2='
Σ {y'd2=' — z'd'y' } =E.m. (y'Z – z'Y).


Thus the equations that determine the motions of a system of bodies are the same, whether the centre of gravity be at rest, or moving uniformly in a straight line; consequently the principles of Impetus, of Least Action, and of the Conservation of Arcas, exist in either case.

166. Let the effect produced by the motion of the centre of gravity

on the position of the plane for which the areas are a maximum, be now determined.

If x + x, ÿ + y, ≈ + z, be put for x, y, z in equations (26), they will retain the same form, namely,

Em (x'dy' - y'dx') = cdt,

Σm (z'dx' - x'dz') = c'dt,

Em (y'dz'z'dy') = c'dt;

for, in consequence of the rectilinear motion of the origin,

- -
ady yd0, zdīīdz 0, ÿdz-zdy = 0.


And as the position of the plane in question is determined by the constant quantities c, c', and c", it will always remain parallel to itself during the motion of the system; on that account it is called the Invariable Plane.

167. Thus, when there are no foreign forces acting on the system, the centre of gravity either remains at rest, or moves uniformly in a straight line; and if that point be assumed as the origin of the co-ordinates, the principles of the conservation of areas and living forces will exist with regard to it; and the invariable plane, always passing through that point, will remain parallel to itself, and will be carried along with the centre of gravity in the general motion of the system.

On the Motion of a System of Bodies in all possible Mathematical relations between Force and Velocity.

168. In nature, force is proportional to velocity; but as a matter of speculation, La Place has investigated the motions of a system of bodies in every possible relation between these two quantities. It is rather singular that such an hypothesis should involve no contradiction; on the contrary, principles similar to the preservation of impetus, the constancy of areas, the motion of the centre of gravity, and the least action, actually exist.


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