Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg"

Chapter XVIII: Part II: Kinetics (3)

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the standard case being that where the rolling sphere is outside the
fixed surface. The opposite case is obtained by reversing the sign of
a. We have also the geometrical relations

[.x] = (a/c)(qz - ry), [.y] = (a/c)(rx - pz), [.z] = (a/c)(py - gx), (23)

If we eliminate P, Q, R from (22), the resulting equations are
integrable with respect to t; thus

M0a M0a
p = - ---(y[.z] - z[.y]) + [alpha], q = - ---(z[.x] - x[.z]) + [beta],
Cc Cc

M0a
r = - ---(x[.y] - y[.x]) + [gamma], (24)
Cc

where [alpha], [beta], [gamma] are arbitrary constants. Substituting
in (23) we find

/ M0a²\ a / M0a²\ a
( 1 + ---- )[.x] = ---([beta]z - [gamma]y), ( 1 + ---- )[.y] = ---([gamma]x - [alpha]z),
\ C / c \ C / c

/ M0a²\ a
( 1 + ---- )[.z] = ---([alpha]y - [beta]x). (25)
\ C / c

Hence [alpha][.x] + [beta][.y] + [gamma][.z] = 0, or

[alpha]x + [beta]y + [gamma]z = const.; (26)

which shows that the centre of the rolling sphere describes a circle.
If the axis of z be taken normal to the plane of this circle we have
[alpha] = 0, [beta] = 0, and

/ M0a²\ a / M0a²\ a
( 1 + ---- )[.x] = -[gamma]--- y, ( 1 + ----- )[.y] = [gamma]--- x. (27)
\ C / c \ C / c

The solution of these equations is of the type

x = b cos ([sigma][tau] + [epsilon]), y = b sin ([sigma][iota] + [epsilon]), (28)

where b, [epsilon] are arbitrary, and

[gamma]a/c
[sigma]= ---------- (29)
1 + M0a²/C

The circle is described with the constant angular velocity [sigma].

When the gravity of the rolling sphere is to be taken into account the
preceding method is not in general convenient, unless the whole motion
of G is small. As an example of this latter type, suppose that a
sphere is placed on the highest point of a fixed sphere and set
spinning about the vertical diameter with the angular velocity n; it
will appear that under a certain condition the motion of G consequent
on a slight disturbance will be oscillatory. If Oz be drawn vertically
upwards, then in the beginning of the disturbed motion the quantities
x, y, p, q, P, Q will all be small. Hence, omitting terms of the
second order, we find

M0[:x] = P, M0[.y] = Q, R = M0g, \
> (30)
C[.p] = -(M0ga/c)y + aQ, C[.q] = (M0ga/c)x - aP, C[.r] = 0. /

The last equation shows that the component r of the angular velocity
retains (to the first order) the constant value n. The geometrical
relations reduce to

[.x] = aq - (na/c)y, [.y] = -ap + (na/c)x. (31)

Eliminating p, g, P, Q, we obtain the equations

(C + M0a²)[:x] + (Cna/c)y - (M0ga²/c)x = 0, }
(C + M0a²)[:y] - (Cna/c)x - (M0ga²/c)y = 0, } (32)

This has two solutions of the type x + iy = [alpha]e^{i([sigma]t +
[epsilon])}, where [alpha], [epsilon] are arbitrary, and [sigma] is a
root of the quadratic

(C + M0a²)[sigma]² - (Cna/c)[sigma] + M0ga²/c = 0. (34)

n² > (4Mgc/C) (1 + M0a²/C), (35)

both roots are real, and have the same sign as n. The motion of G then
consists of two superposed circular vibrations of the type

x = [alpha] cos ([sigma]t + [epsilon]), y = [alpha] sin ([sigma]t + [epsilon]), (36)

in each of which the direction of revolution is the same as that of
the initial spin of the sphere. It follows therefore that the original
position is stable provided the spin n exceed the limit defined by
(35). The case of a sphere spinning about a vertical axis at the
lowest point of a spherical bowl is obtained by reversing the signs of
[alpha] and c. It appears that this position is always stable.

It is to be remarked, however, that in the first form of the problem
the stability above investigated is practically of a limited or
temporary kind. The slightest frictional forces--such as the
resistance of the air--even if they act in lines through the centre of
the rolling sphere, and so do not directly affect its angular
momentum, will cause the centre gradually to descend in an
ever-widening spiral path.

§ 19. _Free Motion of a Solid._--Before proceeding to further problems of motion under extraneous forces it is convenient to investigate the free motion of a solid relative to its mass-centre O, in the most general case. This is the same as the motion about a fixed point under the action of extraneous forces which have zero moment about that point. The question was first discussed by Euler (1750); the geometrical representation to be given is due to Poinsot (1851).

The kinetic energy T of the motion relative to O will be constant. Now T = ½I[omega]², where [omega] is the angular velocity and I is the moment of inertia about the instantaneous axis. If [rho] be the radius-vector OJ of the momental ellipsoid

Ax² + By² + Cz² = M[epsilon]^4 (1)

drawn in the direction of the instantaneous axis, we have I = M[epsilon]^4/[rho]² (§ 11); hence [omega] varies as [rho]. The locus of J may therefore be taken as the "polhode" (§ 18). Again, the vector which represents the angular momentum with respect to O will be constant in every respect. We have seen (§ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that

2T [rho]
OH = ------- · -------, (2)
[Gamma] [omega]

where [Gamma] is the resultant angular momentum about O. Since [omega] varies as [rho], it follows that OH is constant, and the tangent plane at J is therefore fixed in space. The motion of the body relative to O is therefore completely represented if we imagine the momental ellipsoid at O to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact. The fixed plane is parallel to the invariable plane at O, and the line OH is called the _invariable line_. The trace of the point of contact J on the fixed plane is the "herpolhode."

If p, q, r be the component angular velocities about the principal axes at O, we have

(A²p² + B²q² + C²r²)/[Gamma]² = (Ap² + Bq² + Cr²)/2T, (3)

each side being in fact equal to unity. At a point on the polhode cone x : y : z = p : q : r, and the equation of this cone is therefore

/ [Gamma]²\ / [Gamma]²\ / [Gamma]²\
A²( 1 - -------- )x² + B²( 1 - -------- )y² + C²( 1 - -------- )z² = 0. (4)
\ 2AT / \ 2BT / \ 2CT /

Since 2AT - [Gamma]² = B (A - B)q² + C(A - C)r², it appears that if A > B > C the coefficient of x² in (4) is positive, that of z² is negative, whilst that of y² is positive or negative according as 2BT <> [Gamma]². Hence the polhode cone surrounds the axis of greatest or least moment according as 2BT <> [Gamma]². In the critical case of 2BT = [Gamma]² it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a re-entrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B + C > A, the curve has no points of inflexion. The invariable line OH describes another cone in the body, called the _invariable cone_. At any point of this we have x : y : z = Ap. Bq : Cr, and the equation is therefore

/ [Gamma]²\ / [Gamma]²\ / [Gamma]²\
( 1 - -------- )x² + ( 1 - -------- )y² + ( 1 - -------- )z² = 0. (5)
\ 2AT / \ 2BT / \ 2CT /

The signs of the coefficients follow the same rule as in the case of (4). The possible forms of the invariable cone are indicated in fig. 80 by means of the intersections with a concentric spherical surface. In the critical case of 2BT = [Gamma]² the cone degenerates into two planes. It appears that if the body be sightly disturbed from a state of rotation about the principal axis of greatest or least moment, the invariable cone will closely surround this axis, which will therefore never deviate far from the invariable line. If, on the other hand, the body be slightly disturbed from a state of rotation about the mean axis a wide deviation will take place. Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable. The question is greatly simplified when two of the principal moments are equal, say A = B. The polhode and herpolhode cones are then right circular, and the motion is "precessional" according to the definition of § 18. If [alpha] be the inclination of the instantaneous axis to the axis of symmetry, [beta] the inclination of the latter axis to the invariable line, we have

[Gamma] cos [beta] = C [omega] cos [alpha], [Gamma] sin [beta] = A [omega] sin [alpha], (6)

whence

A
tan [beta] = --- tan [alpha]. (7)
C

Hence [beta] <> [alpha], and the circumstances are therefore those of the first or second case in fig. 78, according as A <> C. If [psi] be the rate at which the plane HOJ revolves about OH, we have

sin [alpha] C cos [alpha]
[psi] = ----------- [omega] = ------------- [omega], (8)
sin [beta] A cos [beta]

by § 18 (3). Also if [.chi] be the rate at which J describes the polhode, we have [.psi] sin ([beta]-[alpha]) = [.chi] sin [beta], whence

sin([alpha] - [beta])
[.chi] = --------------------- [omega]. (9)
sin[alpha]

If the instantaneous axis only deviate slightly from the axis of symmetry the angles [alpha], [beta] are small, and [.chi] = (A - C)A·[omega]; the instantaneous axis therefore completes its revolution in the body in the period

2[pi] A - C
------ = ----- [omega]. (10)
[.chi] A

In the case of the earth it is inferred from the independent
phenomenon of luni-solar precession that (C - A)/A = .00313. Hence if
the earth's axis of rotation deviates slightly from the axis of
figure, it should describe a cone about the latter in 320 sidereal
days. This would cause a periodic variation in the latitude of any
place on the earth's surface, as determined by astronomical methods.
There appears to be evidence of a slight periodic variation of
latitude, but the period would seem to be about fourteen months. The
discrepancy is attributed to a defect of rigidity in the earth. The
phenomenon is known as the _Eulerian nutation_, since it is supposed
to come under the free rotations first discussed by Euler.

§ 20. _Motion of a Solid of Revolution._--In the case of a solid of revolution, or (more generally) whenever there is kinetic symmetry about an axis through the mass-centre, or through a fixed point O, a number of interesting problems can be treated almost directly from first principles. It frequently happens that the extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant. For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.

Suppose, for example, that a flywheel is rotating with angular
velocity n about its axis, which is (say) horizontal, and that this
axis is made to rotate with the angular velocity [psi] in the
horizontal plane. The components of angular momentum about the axis of
the flywheel and about the vertical will be Cn and A [psi]
respectively, where A is the moment of inertia about any axis through
the mass-centre (or through the fixed point O) perpendicular to that
of symmetry. If [->OK] be the vector representing the former component
at time t, the vector which represents it at time t + [delta]t will be
[->OK´], equal to [->OK] in magnitude and making with it an angle
[delta][psi]. Hence [->KK´] ( = Cn [delta][psi]) will represent the
change in this component due to the extraneous forces. Hence, so far
as this component is concerned, the extraneous forces must supply a
couple of moment Cn[.psi] in a vertical plane through the axis of the
flywheel. If this couple be absent, the axis will be tilted out of the
horizontal plane in such a sense that the direction of the spin n
approximates to that of the azimuthal rotation [.psi]. The remaining
constituent of the extraneous forces is a couple A[:psi] about the
vertical; this vanishes if [.psi] is constant. If the axis of the
flywheel make an angle [theta] with the vertical, it is seen in like
manner that the required couple in the vertical plane through the axis
is Cn sin [theta] [.psi]. This matter can be strikingly illustrated
with an ordinary gyroscope, e.g. by making the larger movable ring in
fig. 37 rotate about its vertical diameter.

If the direction of the axis of kinetic symmetry be specified by means of the angular co-ordinates [theta], [psi] of § 7, then considering the component velocities of the point C in fig. 83, which are [.theta] and sin [theta][.psi] along and perpendicular to the meridian ZC, we see that the component angular velocities about the lines OA´, OB´ are -sin [theta] [.psi] and [.theta] respectively. Hence if the principal moments of inertia at O be A, A, C, and if n be the constant angular velocity about the axis OC, the kinetic energy is given by

2T = A ([.theta]² + sin² [theta][.psi]²) + Cn². (1)

Again, the components of angular momentum about OC, OA´ are Cn, -A sin [theta] [.psi], and therefore the angular momentum ([mu], say) about OZ is

[mu] = A sin² [theta][.psi] + Cn cos [theta]. (2)

We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the direction OG. If [theta] is constant the points C, A´ will in time [delta]t come to positions C´´, A´´ such that CC´´ = sin [theta] [delta][psi], A´A´´ = cos [theta] [delta][psi], and the angular momentum about OB´ will become Cn sin [theta] [delta][psi] - A sin [theta] [.psi] · cos [theta] [delta][psi]. Equating this to Mgh sin [theta] [delta]t, and dividing out by sin [theta], we obtain

A cos [theta] [.psi]² - Cn[.psi] + Mgh = 0, (3)

as the condition in question. For given values of n and [theta] we have two possible values of [.psi] provided n exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with [root](4AMgh cos [theta]), one value of [.psi] is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos [theta] approximately. The absence of g from the latter expression indicates that the circumstances of the rapid precession are very nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.

Again, take the case of a circular disk rolling in steady motion on a
horizontal plane. The centre O of the disk is supposed to describe a
horizontal circle of radius c with the constant angular velocity
[.psi], whilst its plane preserves a constant inclination [theta] to
the horizontal. The components of the reaction of the horizontal lane
will be Mc[.psi]² at right angles to the tangent line at the point of
contact and Mg vertically upwards, and the moment of these about the
horizontal diameter of the disk, which corresponds to OB´ in fig. 83,
is Mc[.psi]². [alpha] sin [theta] - Mg[alpha] cos [theta], where
[alpha] is the radius of the disk. Equating this to the rate of
increase of the angular momentum about OB´, investigated as above, we
find

/ a \ a²
( C + Ma² + A --- cos [theta] ) [.psi]² = Mg --- cot [theta], (4)
\ c / c

where use has been made of the obvious relation n[alpha] = c[.psi]. If
c and [theta] be given this formula determines the value of [psi] for
which the motion will be steady.

In the case of the top, the equation of energy and the condition of constant angular momentum ([mu]) about the vertical OZ are sufficient to determine the motion of the axis. Thus, we have

½A ([.theta]² + sin² [theta][.psi]²) + ½Cn² + Mgh cos [theta] = const., (5)

A sin² [theta][.psi] + [nu] cos [theta] = [mu], (6)

where [nu] is written for Cn. From these [.psi] may be eliminated, and on differentiating the resulting equation with respect to t we obtain

([mu] - [nu] cos [theta])([mu] cos [theta] - [nu])
A[:theta] - -------------------------------------------------- - Mgh sin [theta] = 0. (7)
A sin³ [theta]

If we put [:theta] = 0 we get the condition of steady precessional motion in a form equivalent to (3). To find the small oscillation about a state of steady precession in which the axis makes a constant angle [alpha] with the vertical, we write [theta] = [alpha] + [chi], and neglect terms of the second order in [chi]. The result is of the form

[:chi] + [sigma]²[chi] = 0, (8)

where

[sigma]² = {([mu] - [nu] cos [alpha])² + 2([mu] - [nu] cos [alpha])([mu] cos [alpha] - [nu])
cos [alpha] + ([mu] cos [alpha] - [nu])²} / A² sin^4 [alpha]. (9)

When [nu] is large we have, for the "slow" precession [sigma] = [nu]/A, and for the "rapid" precession [sigma] = A/[nu] cos [alpha] = [.psi], approximately. Further, on examining the small variation in [.psi], it appears that in a slightly disturbed slow precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be easily observed. In a slightly disturbed rapid precession the superposed vibration is elliptic-harmonic, with a period equal to that of the precession itself. The ratio of the axes of the ellipse is sec [alpha], the longer axis being in the plane of [theta]. The result is that the axis of the top describes a circular cone about a fixed line making a small angle with the vertical. This is, in fact, the "invariable line" of the free Eulerian rotation with which (as already remarked) we are here virtually concerned. For the more general discussion of the motion of a top see GYROSCOPE.

§ 21. _Moving Axes of Reference._--For the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.

We will assume for the present that the origin O is fixed. The moving axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The components of angular momentum about Ox, Oy, Oz will be denoted as usual by [lambda], [mu], [nu]. Now consider a system of fixed axes Ox´, Oy´, Oz´ chosen so as to coincide at the instant t with the moving system Ox, Oy, Oz. At the instant t + [delta]t, Ox, Oy, Oz will no longer coincide with Ox´, Oy´, Oz´; in particular they will make with Ox´ angles whose cosines are, to the first order, 1, -r[delta]t, q[delta]t, respectively. Hence the altered angular momentum about Ox´ will be [lambda] + [delta][lambda] + ([mu] + [delta][mu]) (-r[delta]t) + ([nu] + [delta][nu]) q[delta]t. If L, M, N be the moments of the extraneous forces about Ox, Oy, Oz this must be equal to [lambda] + L[delta]t. Hence, and by symmetry, we obtain

d[lambda] \
--------- - r[nu] + q[nu] = L, |
dt |
|
d[mu] |
----- - p[nu] + r[lanbda] = M, > (1)
dt |
|
d[nu] |
----- - q[lambda] + p[nu] = N. |
dt /

These equations are applicable to any dynamical system whatever. If we now apply them to the case of a rigid body moving about a fixed point O, and make Ox, Oy, Oz coincide with the principal axes of inertia at O, we have [lambda], [mu], [nu] = Ap, Bq, Cr, whence

dp \
A -- - (B - C) qr = L, |
dt |
|
dq |
B -- - (C - A) rp = M, > (2)
dt |
|
dr |
C -- - (A - B) pq = N. |
dt /

If we multiply these by p, q, r and add, we get

d
--- · ½(Ap² + Bq² + Cr²) = Lp + Mq + Nr, (3)
dt

which is (virtually) the equation of energy.

As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity [omega] about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining couple is

L = (C - B) mn[omega]², M = (A - C) nl[omega]², N = (B - A) lm[omega]². (4)

If we reverse the signs, we get the "centrifugal couple" exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).

If in (2) we put, L, M, N = O we get the case of free rotation; thus

dp \
A -- = (B - C) qr, |
dt |
|
dq |
B -- = (C - A) rp, > (5)
dt |
|
dr |
C -- = (A - B) pq. |
dt /

These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap² + Bq² + Cr² and A²p² + B²q² + C²r² are both constant. The former is, in fact, equal to 2T, and the latter to [Gamma]², where T is the kinetic energy and [Gamma] the resultant angular momentum.

To complete the solution of (2) a third integral is required; this
involves in general the use of elliptic functions. The problem has
been the subject of numerous memoirs; we will here notice only the
form of solution given by Rueb (1834), and at a later period by G.
Kirchhoff (1875), If we write
_
/ [phi] d[phi]
u = | ------------, [Delta][phi] = [root](1 - k² sin² [phi]),
_/ 0 [Delta][phi]

we have, in the notation of elliptic functions, [phi] = am u. If we
assume

p = p0 cos am ([sigma]t + [epsilon]), q = q0sin am ([sigma]t + [epsilon]),
r = r0[Delta] am ([sigma]t + [epsilon]), (7)

we find

[sigma]p0 [sigma]q0 k²[sigma]r0
[.p] = - --------- qr, [.q] = --------- rp, [.r] = - ----------- pq. (8)
q0r0 r0p0 p0q0

Hence (5) will be satisfied, provided

-[sigma]p0 B - C [sigma]q0 C - A -k²[sigma]r0 A - B
---------- = -----, --------- = -----, ------------ = -----. (9)
q0r0 A r0p0 B p0q0 C

These equations, together with the arbitrary initial values of p, q,
r, determine the six constants which we have denoted by p0, q0, r0,
k², [sigma], [epsilon]. We will suppose that A > B > C. From the form
of the polhode curves referred to in § 19 it appears that the angular
velocity q about the axis of mean moment must vanish periodically. If
we adopt one of these epochs as the origin of t, we have [epsilon] =
0, and p0, r0 will become identical with the initial values of p, r.
The conditions (9) then lead to

A(A - C) (A - C)(B - C) A(A - B) p0²
q0² = -------- p0², [sigma]² = -------------- r0², k² = -------- · ---. (10)
B(B - C) AB C(B - C) r0²

For a real solution we must have k² < 1, which is equivalent to 2BT > [Gamma]². If the initial
conditions are such as to make 2BT < [Gamma]², we must interchange the
forms of p and r in (7). In the present case the instantaneous axis
returns to its initial position in the body whenever [phi] increases
by 2[pi], i.e. whenever t increases by 4K/[sigma], when K is the
"complete" elliptic integral of the first kind with respect to the
modulus k.

The elliptic functions degenerate into simpler forms when k² = 0 or k²
= 1. The former case arises when two of the principal moments are
equal; this has been sufficiently dealt with in § 19. If k² = 1, we
must have 2BT = [Gamma]². We have seen that the alternative 2BT <>
[Gamma]² determines whether the polhode cone surrounds the principal
axis of least or greatest moment. The case of 2BT = [Gamma]², exactly,
is therefore a critical case; it may be shown that the instantaneous
axis either coincides permanently with the axis of mean moment or
approaches it asymptotically.

When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are

d[xi] d[eta] d[zeta]
----- - r[eta] + q[zeta] = X, ------ - p[zeta] - r[chi] = Y, ------- - q[chi] + p[eta] = Z, (11)
dt dt dt

d[lambda] d[mu] \
--------- - r[mu] + q[nu] - w[eta] + v[zeta] = L, ----- - p[nu] + r[lambda]- u[zeta] + w[xi] = M, |
dt dt |
> (12)
d[nu] |
----- - q[lambda] + p[mu] - v[xi] + u[eta] = N. /
dt

To prove these, we may take fixed axes O´x´, O´y´, O´z´ coincident with the moving axes at time t, and compare the linear and angular momenta [xi] + [delta][xi], [eta] + [delta][eta], [zeta] + [delta][zeta], [lambda] + [delta][lambda], [mu] + [delta][mu], [nu] + [delta][nu] relative to the new position of the axes, Ox, Oy, Oz at time t + [delta]t with the original momenta [xi], [eta], [zeta], [lambda], [mu], [nu] relative to O´x´, O´y´, O´z´ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the moving origin coincide always with the mass-centre, we have [xi], [eta], [zeta] = M0u, M0v, M0w, where M0 is the total mass, and the equations simplify.

When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of motion about a fixed point O; the angular co-ordinates [theta], [phi], [psi] of Euler may then be used for the purpose. Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their instantaneous positions are

p = [.theta] sin [phi] - sin [theta] cos [phi][.psi], \
q = [.theta] cos [phi] + sin [theta] sin [phi][.psi], > (13)
r = [.phi] + cos [theta][.psi], /

by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by

2T = A([.theta] sin [phi] - sin [theta] cos [phi][.psi])² \
+ B([.theta] cos [phi] + sin [theta] sin [theta][psi])² > (14)
+ C([.phi] + cos [theta][.psi])². /

If A = B this reduces to

2T = A([.theta]² + sin² [theta][.psi]²) + C([.phi] + cos [theta][.psi])²; (15)

cf. § 20 (1).

§ 22. _Equations of Motion in Generalized Co-ordinates._--Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but their number is determinate, and expresses the number of _degrees of freedom_ of the system. We denote these co-ordinates by q1, q2, ... q_n. It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the q's, varying in form (of course) from particle to particle. Hence the kinetic energy T is given by

__
2T = \ {m([.x]² + [.y]² + [.z]²)}
/__

= a11[.q]1² + a22[.q]2² + ... + 2a12[.q]1[.q]2 + ..., (1)

where
_ _
__ | { / [dP]x \² / [dP]y \² / [dP]z \² } | \
a_rr = \ | m { ( ------- ) + ( ------- ) + ( ------- ) } |, |
/__ |_ { \[dP]q_r/ \[dP]q_r/ \[dP]q_r/ } _| |
_ _ > (2)
__ | / [dP]x [dP]x [dP]y [dP]y [dP]z [dP]z \ | |
a_rs = \ | m ( ------- ------- + ------- ------- + ------- ------- ) | = a_sr. |
/__ |_ \[dP]q_r [dP]q_s [dP]q_r [dP]q_s [dP]q_r [dP]q_s/ _| /

Thus T is expressed as a homogeneous quadratic function of the quantities [.q]1, [.q]2, ... [.q]_n, which are called the _generalized components of velocity_. The coefficients a_rr, a_rs are called the coefficients of inertia; they are not in general constants, being functions of the q's and so variable with the configuration. Again, If (X, Y, Z) be the force on m, the work done in an infinitesimal change of configuration is

[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ... + Q_n[delta]q_n, (3)

where

/ [dP]x [dP]y [dP]z \
Q_r = [Sigma]( X------- + Y------- + Z------- ). (4)
\ [dP]q_r [dP]q_r [dP]q_r /

The quantities Q_r are called the _generalized components of force_.

The equations of motion of m being

m[:x] = X, m[:y] = Y, m[:z] = Z, (5)

Now

[dP]x [dP]x [dP]x
[.x] = ------[.q]1 + ------[.q]2 + ... + -------[.q]_n, (7)
[dP]q1 [dP]q2 [dP]q_n

whence

[dP][.x] [dP]x
---------- = -------. (8)
[dP][.q]_r [dP]q_r

Also

d / [dP]x \ [dP]²x [dP]²x [dP]²x [dP]x
-- ( ------- ) = ------------[.q]1 + -------------[.q]2 + ... + --------------[.q]_r = --------. (9)
dt \[dP]q_r/ [dP]q1[dP]q_r [dP]q2[dP]q_r [dP]q_n[dP]q_r [dP]q_r

Hence

[dP]x d / [dP]x \ d / [dP]x \ d / [dP][.x] \ [dP][.x]
[:x]------- = ---( [.x]------- ) - [.x]---( ------- ) = ---( [.x]---------- ) - [.x]--------. (10)
[dP]q_r dt \ [dP]q_r/ dt \[dP]q_r/ dt \ [dP][.q]_r/ [dP]q_r

By these and the similar transformations relating to y and z the equation (6) takes the form

d / [dP]T \ [dP]T
--- ( ---------- ) - ------ = Q_r. (11)
dt \[dP][.q]_r/ [dP]q_r

If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated. The above proof was given by Sir W. R. Hamilton (1835). Lagrange's own proof will be found under DYNAMICS, § _Analytical_. In a conservative system free from extraneous force we have

[Sigma](X [delta]x + Y [delta]y + Z [delta]z) = -[delta]V, (12)

where V is the potential energy. Hence

[dP]V
Q_r = - -------, (13)
[dP]q_r

and

d / [dP]T \ [dP]T [dP]V
--- ( ---------- ) - ----- = - -------. (14)
dt \[dP][.q]_r/ Vq_r [dP]q_r

If we imagine any given state of motion ([.q]1, [.q]2 ... [.q]_n) through the configuration (q1, q2, ... q_n) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11) with respect to t over the infinitely short duration of the impulse

[dP]T
---------- = Q_r´, (15)
[dP][.q]_r

where Q_r´ is the time integral of Q_r and so represents a _generalized component of impulse_. By an obvious analogy, the expressions [dP]T/[dP][.q]_r may be called the _generalized components of momentum_; they are usually denoted by p_r thus

p_r = [dP]T/[dP][.q]_r = a_(1r)[.q]1 + a_(2r)[.q]2 + ... + a_(nr)[.q]_n. (16)

Since T is a homogeneous quadratic function of the velocities [.q]1, [.q]2, ... [.q]_n, we have

[dP]T [dP]T [dP]T
2T = ---------[.q]1 + ---------[.q]2 + ... + ----------[.q]_n = p1[.q]2 + p2[.q]2 + ... + p_n[.q]_n. (17)
[dP][.q]1 [dP][.q]2 [dP][.q]_n

Hence

dT
2-- = [.p]1[.q]1 + [.p]2[.q]2 + ... [.p]_n[.q]_n \
dt |
|
+ [.p]1[:q]1 + [.p]2[:q]2 + ... + [.p]_n[:q]_n |
|
/ [dP]T \ / [dP]T \ / [dP]T \ |
= ( --------- + Q1 ) [.q]1 + ( --------- + Q2 ) [.q]2 + ... + ( ---------- + Q_n )[.q]_n > (18)
\[dP][.q]1 / \[dP][.q]2 / \[dP][.q]_n / |
|
[dP]T [dP]T [dP]T |
+ ---------[:q]1 + ---------[:q]2 + ... ----------[:q]_n |
[dP][.q]1 [dP][.q]2 [dP][.q]_n |
|
dT |
= -- + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n, /
dt

dT
-- = Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (19)
dt

This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces. In the case of a conservative system free from extraneous force it becomes the equation of energy

d
--- (T + V) = 0, or T + V = const., (20)
dt

in virtue of (13).

As a first application of Lagrange's formula (11) we may form the
equations of motion of a particle in spherical polar co-ordinates. Let
r be the distance of a point P from a fixed origin O, [theta] the
angle which OP makes with a fixed direction OZ, [psi] the azimuth of
the plane ZOP relative to some fixed plane through OZ. The
displacements of P due to small variations of these co-ordinates are
[dP]r along OP, r [delta][theta] perpendicular to OP in the plane ZOP,
and r sin [theta] [delta][psi] perpendicular to this plane. The
component velocities in these directions are therefore [.r],
r[.theta], r sin [theta][.psi], and if m be the mass of a moving
particle at P we have

2T = m([.r]² + r²[.theta]² + r² sin² [theta][.psi]²). (21)

Hence the formula (11) gives

The quantities R, [Theta], [Psi] are the coefficients in the
expression R [delta]r + [Theta] [delta][theta] + [Psi] [delta][psi]
for the work done in an infinitely small displacement; viz. R is the
radial component of force, [Theta] is the moment about a line through
O perpendicular to the plane ZOP, and [Psi] is the moment about OZ. In
the case of the spherical pendulum we have r = l, [Theta] = - mgl sin
[theta], [Psi] = 0, if OZ be drawn vertically downwards, and therefore

g \
[:theta] - sin [theta] cos [theta][.psi]² = - --- sin [theta], |
l > (23)
|
sin² [theta][.psi] = h, /

where h is a constant. The latter equation expresses that the angular
momentum ml² sin² [theta][.psi] about the vertical OZ is constant. By
elimination of [.psi] we obtain

g
[:theta] - h² cos² [theta] / sin^3[theta] = - --- sin [theta]. (24)
l

If the particle describes a horizontal circle of angular radius
[alpha] with constant angular velocity [Omega], we have [.omega] = 0,
h = [Omega]² sin [alpha], and therefore

g
[Omega]² = --- cos [alpha], (25)
l

as is otherwise evident from the elementary theory of uniform circular
motion. To investigate the small oscillations about this state of
steady motion we write [theta] = [alpha] + [chi] in (24) and neglect
terms of the second order in [chi]. We find, after some reductions,

[:chi] + (1 + 3 cos² [alpha]) [Omega]²[chi] = 0; (26)

this shows that the variation of [chi] is simple-harmonic, with the
period

2[pi]/[root](1 + 3 cos² [alpha])·[Omega]

As regards the most general motion of a spherical pendulum, it is
obvious that a particle moving under gravity on a smooth sphere cannot
pass through the highest or lowest point unless it describes a
vertical circle. In all other cases there must be an upper and a lower
limit to the altitude. Again, a vertical plane passing through O and a
point where the motion is horizontal is evidently a plane of symmetry
as regards the path. Hence the path will be confined between two
horizontal circles which it touches alternately, and the direction of
motion is never horizontal except at these circles. In the case of
disturbed steady motion, just considered, these circles are nearly
coincident. When both are near the lowest point the horizontal
projection of the path is approximately an ellipse, as shown in § 13;
a closer investigation shows that the ellipse is to be regarded as
revolving about its centre with the angular velocity 2/3 ab[Omega]/l²,
where a, b are the semi-axes.

To apply the equations (11) to the case of the top we start with the
expression (15) of § 21 for the kinetic energy, the simplified form
(1) of § 20 being for the present purpose inadmissible, since it is
essential that the generalized co-ordinates employed should be
competent to specify the position of every particle. If [lambda],
[mu], [nu] be the components of momentum, we have

The meaning of these quantities is easily recognized; thus [lambda] is
the angular momentum about a horizontal axis normal to the plane of
[theta], [mu] is the angular momentum about the vertical OZ, and [nu]
is the angular momentum about the axis of symmetry. If M be the total
mass, the potential energy is V = Mgh cos [theta], if OZ be drawn
vertically upwards. Hence the equations (11) become

A[:theta] - A sin [theta] cos [theta][.psi]² + C([.phi] + cos [theta][.psi]) [.psi] sin [theta] = Mgh sin [theta], \
d/dt · {A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta]} = 0, > (28)
d/dt · {C([.phi] + cos [theta][.psi])} = 0, /

of which the last two express the constancy of the momenta [mu], [nu].
Hence

A[:theta] - A sin [theta] cos [theta][.psi]² + [nu] sin [theta][.psi] = Mgh sin [theta], \ (29)
A sin² [theta][.psi] + [nu] cos [theta] = [mu]. /

If we eliminate [.psi] we obtain the equation (7) of § 20. The theory
of disturbed precessional motion there outlined does not give a
convenient view of the oscillations of the axis about the vertical
position. If [theta] be small the equations (29) may be written

[nu]²- 4AMgh \
[:theta] - [theta][.omega]² = - ------------[theta], > (30)
4A² |
[theta]²[.omega] = const., /

where

[nu]
[omega] = [psi] - ---- t. (31)
2A

Since [theta], [omega] are the polar co-ordinates (in a horizontal
plane) of a point on the axis of symmetry, relative to an initial line
which revolves with constant angular velocity [nu]/2A, we see by
comparison with § 14 (15) (16) that the motion of such a point will be
elliptic-harmonic superposed on a uniform rotation [nu]/2A, provided
[nu]² > 4AMgh. This gives (in essentials) the theory of the
"gyroscopic pendulum."

§ 23. _Stability of Equilibrium. Theory of Vibrations._--If, in a conservative system, the configuration (q1, q2, ... q_n) be one of equilibrium, the equations (14) of § 22 must be satisfied by [.q]1, [.q]2 ... [.q]_n = 0, whence

[dP]V / [dP]q_r = 0. (1)

A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance. No such simple proof is available to show without qualification that the above condition is _necessary_. If, however, we recognize the existence of dissipative forces called into play by any motion whatever of the system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V must continually diminish so long as the velocities [.q]1, [.q]2, ... [.q]_n differ from zero. Hence if the system be started from rest in a configuration for which V is less than in the equilibrium configuration considered, this quantity must still further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).

In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized cc-ordinates q1, q2, ... q_n that they shall vanish in the configuration in question. The potential energy is then given with sufficient approximation by an expression of the form

2V = c11q1² + c22q2² + ... + 2c12q1q2 + ..., (2)

a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V is stationary. The coefficients c_rr, c_rs are called _coefficients of stability_. We may further treat the coefficients of inertia a_rr, a_rs of § 22 (1) as constants. The Lagrangian equations of motion are then of the type

a_(1r)[:q]1 + a_(2r)[:q]2 + ... + a_(nr)[:q]_n + c_(1r)q1 + c_(2r)q2 + ... + c_(nr)q_n = Q_r, (3)

where Q_r now stands for a component of extraneous force. In a _free oscillation_ we have Q1, Q2, ... Q_n = 0, and if we assume

q_r = A_r e^(i[sigma]^t), (4)

we obtain n equations of the type

(c_(1r) - [sigma]²a_(1r)) A1 + (c_(2r) - [sigma]²a_(2r)) A2 + ... + (c_(nr) - [sigma]²a_nr) A_n = 0. (5)

Eliminating the n - 1 ratios A1 : A2 : ... : A_n we obtain the determinantal equation

[Delta]([sigma]²) = 0, (6)

where

[Delta]([sigma]²) = | c11 - [sigma]²a11, c21 - [sigma]²a21, ..., C_(n1) - [sigma]²a_(nl) |
| c12 - [sigma]²a12, c22 - [sigma]²a22, ..., C_(n2) - [sigma]²a_(n2) |
| . . ... . |
| . . ... . | (7)
| . . ... . |
| c_(1n) - [sigma]²a{1n}, c_(2n) - [sigma]²a_(2n), ..., C_(nn) - [sigma]²a_(nn) |

The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation in [sigma]² are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities A1, A2, ... A_n, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors of any one row in the determinate [Delta]([sigma]²). By combining the solutions corresponding to a pair of equal and opposite values of [sigma] we obtain a solution in real form:

q_r = C_(a_r) cos ([sigma]t + [epsilon]), (8)

where a1, a2 ... a_r are a determinate series of quantities having to one another the above-mentioned ratios, whilst the constants C, [epsilon] are arbitrary. This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the q's) executes a simple vibration of period 2[pi]/[sigma]. The amplitudes of oscillation of the various particles have definite ratios to one another, and the phases are in agreement, the absolute amplitude (depending on C) and the phase-constant ([epsilon]) being alone arbitrary. A vibration of this character is called a _normal mode_ of vibration of the system; the number n of such modes is equal to that of the degrees of freedom possessed by the system. These statements require some modification when two or more of the roots of the equation (6) are equal. In the case of a multiple root the minors of [Delta]([sigma]²) all vanish, and the basis for the determination of the quantities a_r disappears. Two or more normal modes then become to some extent indeterminate, and elliptic vibrations of the individual particles are possible. An example is furnished by the spherical pendulum (§ 13).

As an example of the method of determination of the normal modes we
may take the "double pendulum." A mass M hangs from a fixed point by a
string of length a, and a second mass m hangs from M by a string of
length b. For simplicity we will suppose that the motion is confined
to one vertical plane. If [theta], [phi] be the inclinations of the
two strings to the vertical, we have, approximately,

2T = Ma²[.theta]² + m(a[.theta] + b[.psi])² \ (9)
2V = Mga[theta]² + mg(a[theta]² + b[psi]²). /

The equations (3) take the forms

a[:theta] + [mu]b[:phi] + g[theta] = 0, \ (10)
a[:theta] + b[:phi] + g[phi] = 0. /

where [mu] = m/(M + m). Hence

([sigma]² - g/a)a[theta] + [mu][sigma]²b[phi] = 0, \ (11)
[sigma]²a[theta] + ([sigma]² - g/b)b[phi] = 0. /

The frequency equation is therefore

([sigma]² - g/a)([sigma]² - g/b) - [mu][sigma]^4 = 0. (12)

The roots of this quadratic in [sigma]² are easily seen to be real and
positive. If M be large compared with m, [mu] is small, and the roots
are g/a and g/b, approximately. In the normal mode corresponding to
the former root, M swings almost like the bob of a simple pendulum of
length a, being comparatively uninfluenced by the presence of m,
whilst m executes a "forced" vibration (§ 12) of the corresponding
period. In the second mode, M is nearly at rest [as appears from the
second of equations (11)], whilst m swings almost like the bob of a
simple pendulum of length b. Whatever the ratio M/m, the two values of
[sigma]² can never be exactly equal, but they are approximately equal
if a, b are nearly equal and [mu] is very small. A curious phenomenon
is then to be observed; the motion of each particle, being made up (in
general) of two superposed simple vibrations of nearly equal period,
is seen to fluctuate greatly in extent, and if the amplitudes be equal
we have periods of approximate rest, as in the case of "beats" in
acoustics. The vibration then appears to be transferred alternately
from m to M at regular intervals. If, on the other hand, M is small
compared with m, [mu] is nearly equal to unity, and the roots of (12)
are [sigma]² = g/(a + b) and [sigma]² = mg/M·(a + b)/ab,
approximately. The former root makes [theta] = [phi], nearly; in the
corresponding normal mode m oscillates like the bob of a simple
pendulum of length a + b. In the second mode a[theta] + b[phi] = 0,
nearly, so that m is approximately at rest. The oscillation of M then
resembles that of a particle at a distance a from one end of a string
of length a + b fixed at the ends and subject to a tension mg.

The motion of the system consequent on arbitrary initial conditions may be obtained by superposition of the n normal modes with suitable amplitudes and phases. We have then

q_r = [alpha]_r[theta] + [alpha]_r´[theta]´ + [alpha]_r´´[theta]´´ + ..., (13)

where

[theta] = C cos ([sigma]t + [epsilon]), [theta]´
= C´ cos ([sigma]´t + [epsilon]), [theta]´´
= C´´ cos([sigma]´´t + [epsilon]), ... (14)

provided [sigma]², [sigma]´², [sigma]´´², ... are the n roots of (6). The coefficients of [theta], [theta]´, [theta]´´, ... in (13) satisfy the _conjugate_ or _orthogonal_ relations

a11[alpha]1[alpha]1´ + a22[alpha]2[alpha]2´ + ... + a12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0, (15)
c11[alpha]1[alpha]1´ + c22[alpha]2[alpha]2´ + ... + c12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0, (16)

provided the symbols [alpha]_r, [alpha]_r´ correspond to two distinct roots [sigma]², [sigma]´² of (6). To prove these relations, we replace the symbols A1, A2, ... A_n in (5) by [alpha]1, [alpha]2, ... [alpha]_n respectively, multiply the resulting equations by a´1, a´2, ... a´_n, in order, and add. The result, owing to its symmetry, must still hold if we interchange accented and unaccented Greek letters, and by comparison we deduce (15) and (16), provided [sigma]² and [sigma]´² are unequal. The actual determination of C, C´, C´´, ... and [epsilon], [epsilon]´, [epsilon]´´, ... in terms of the initial conditions is as follows. If we write

C cos [epsilon] = H, -C sin [epsilon] = K, (17)

we must have

[alpha]_rH + [alpha]_r´H´ + [alpha]_r´´H´´ + ... = [q_r]0, \ (18)
[sigma][alpha]_rH + [sigma]´[alpha]_r´H´ + [sigma]´´[alpha]_r´´H´´ + ... = [[.q]_r]0, /

where the zero suffix indicates initial values. These equations can be at once solved for H, H´, H´´, ... and K, K´, K´´, ... by means of the orthogonal relations (15).

By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to sums of squares. The transformation is in fact effected by the assumption (13), in virtue of the relations (15) (16), and we may write

2T = a[.theta]² + a´[.theta]´² + a´´[.theta]´´² + ..., \ (19)
2V = c[theta]² + c´[theta]´² + c´´[theta]´´² + .... /

The new co-ordinates [theta], [theta]´, [theta]´´ ... are called the _normal_ co-ordinates of the system; in a normal mode of vibration one of these varies alone. The physical characteristics of a normal mode are that an impulse of a particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a particular normal type maintains a displacement of that type only. The normal modes are further distinguished by an important "stationary" property, as regards the frequency. If we imagine the system reduced by frictionless constraints to one degree of freedom, so that the co-ordinates [theta], [theta]´, [theta]´´, ... have prescribed ratios to one another, we have, from (19),

c[theta]² + c´[theta]´² = c´´[theta]´´² + ...
[sigma]² = ---------------------------------------------, (20)
a[theta]² + a´[theta]´² + a´´[theta]´´² + ...

This shows that the value of [sigma]² for the constrained mode is intermediate to the greatest and least of the values c/a, c´/a´, c´´/a´´, ... proper to the several normal modes. Also that if the constrained mode differs little from a normal mode of free vibration (e.g. if [theta]´, [theta]´´, ... are small compared with [theta]), the change in the frequency is of the second order. This property can often be utilized to estimate the frequency of the gravest normal mode of a system, by means of an assumed approximate type, when the exact determination would be difficult. It also appears that an estimate thus obtained is necessarily too high.

From another point of view it is easily recognized that the equations (5) are exactly those to which we are led in the ordinary process of finding the stationary values of the function

V (q1, q2, ... q_n)
------------------------,
T (q1, q2, ... q_n)

where the denominator stands for the same homogeneous quadratic function of the q's that T is for the [.q]'s. It is easy to construct in this connexion a proof that the n values of [sigma]² are all real and positive.

The case of three degrees of freedom is instructive on account of the
geometrical analogies. With a view to these we may write

2T= a[.x]² + b[.y]² + c[.z]² + 2f[.y][.z] + 2g[.z][.x] + 2h[.x][.y], \ (21)
2V = Ax² + By² + Cz² + 2Fyz + 2Gzx + 2Hxy. /

It is obvious that the ratio

V (x, y, z)
----------- (22)
T (x, y, z)

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Encyclopaedia Britannica, 11th Edition, "Matter" to "Mecklenburg"Chapter XVIII: Part II: Kinetics (3)

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