so that we may write •
A (dy\ _ d8y I dx _ d£ dSx /dx\*
\dx) dt I ~dt ~di ~dt~ \di)
or, once more removing t from explicit appearance,
* /dy\ _ ddy dy ddx
\dx) dx dx dx
If x is the independent variable, dx = 0, so that we have the same
formula as before.)
Let us now find the variation of the integral
, x, y, z, x', y1, /, . . .) dt.
Changing x to x + dx, y to y -f dy, x' to x1 + dx', etc.,
7+ dl+ (?2/ + - =< -f d 4-
and the variations are
- =f(ddx).
It may, on occasion, be more convenient to use these more general
formulae, not supposing the variation of any variable to vanish.)
If the limits are varied, we have, indicating the part of the
change in I due to the change in either limit by a suffix,
*? * r hfh
\I = I cpdt — lcpdt= lcpdt =
J J J
which are to added to the part already found.
In the application of the calculus of variations, we often
encounter problems involving a number of independent variables, so
that we deal with partial derivatives, and multiple integrals. The
principles here given will however suffice for the treatment of all
the usual questions.
As a celebrated mechanical example of the use of the Calculus
of Variations let us consider the question: What is that curve along
which a particle must be constrained to descend under the influence
of gravity in order to pass from one point to another in the least
possible time?
Since v = ^ S.> we have for the time of descent t = I - > or
dt
making use of the equation of energy § 27, 27),
= I -
J v
Let us take for the independent variable corresponding to t above
the vertical coordinate 0. We suppose the motion to take place
in a vertical plane. We have then
If now we make an arbitrary infinitesimal variation of the
curve, if t is to be a minimum we mast have the term of the first
order in s vanish,
dt = 0.
WEBSTER, Dynamics. 6
82 HI. GENERAL PRINCIPLES. WORK AND ENERGY.
Now
t =
•o
For any particular curve x is a given function of 3. Giving it a
variation dx we have
§ x'dx'dz
ot
-f
Making use of dx' = ^--- and integrating by parts1),
Zi Zl
x' dx
de
If the ends of the curve are fixed dx vanishes for both limits #0
and 0i, hence the integrated part vanishes. Consequently for a
minimum the integral must vanish.
Now since the function da; is purely arbitrary if the other factor
of the integrand did not vanish for any points of the curve we
might take dx of the same sign as that factor at each point. Thus
the integrand would be positive everywhere and the integral would
not vanish, consequently the factor multiplying Sx must vanish
for each point of the curve, or
This is the differential equation of the curve of quickest descent,
or brachistochrone.
Integrating we have
x'
- = c, an arbitrary constant.
y(l + *")(V-*0[*-*o])
Squaring and solving for xt2 we obtain
V 2 1 V 2
Let us put a = ~ -f #0, & = ^— ^ — ^ ^0 (6 is arbitrary, since it
involves c), then we have
1) The bar / signifies that we are to subtract the value of the expression
before it at the lower limit z0 from the value at the upper limit z:.
29] BRACHISTOCHRONE FOR GRAVITY.
If we introduce a new variable # such that
— I} a4-b
g --- ~ COS ft,
we have
dz = -s
83
Thus our differential equation becomes
dx -i/l -f cos & 1 -f- cos #
Consequently
Integrating,
1 — cos &
sin #
where J 70
-f I -5 -TS— ) cos (ny) -f I -g -5-7 cos (nz) \ d S
\dz dx) \ox dy] J\
taken over the portion of the surface bounded by the paths 1 and 2
from A to B. Now — I± may be considered the integral from S
to A along the path 1, so that I2 — /x is the integral around the
closed path which forms the contour of the portion of surface S.
We accordingly get the following, known as
STOKES'S THEOREM.1) The line integral, around any closed contour;
of the tangential component of a vector JR, whose components are
X, Y, Z, is equal to the surface integral over any portion of surface
bounded by the contour, of the normal component of a vector w,
whose components £, 77, £ are related to X, F, Z by the relations
t_<^_<>Z
*~ dy dz'
ax az
•J1 = -^ -pr )
02 OX
ar ax
1) The proof here given is from the author's notes on the lectures of
Professor von Helmholtz. A similar treatment is given by Picard, Traite
d' Analyse, Tom. I, p. 73.
30, 31]
STOKES'S THEOREM. CURL.
87
The normal must be drawn toward that side of the surface that
shall make the rotation of a right-handed screw advancing along the
normal agree with the direction of traversing the closed contour of
integration.
37) IE cos (By ds) ds = I Xdx + Ydy + Zdz = I la cos (on) dS
= I I { % cos (nx) + ri cos (ny) + g cos (ni)} dS.
The vector co related to the vector point -function E by the differ-
ential equations above is called the rotation, spin (Clifford), or curl
(Maxwell and Heaviside) of JR. Such
vectors are of frequent occurrence
in mathematical physics. (See
Part III.)
The significance of the geo-
metrical term curl can be seen
from the physical example in
which the vector E represents
the velocity of a point instant-
aneously occupying the position
x, y, z in a rigid body turning
about the ^"-axis with an angular
velocity co. Then the vector E= OQ is perpendicular to the radius
and its components are (Fig. 22),
0
rig
X = B cos (Ex) = — E sin
= BGOS(QX) =
= XG),
where co is constant, and
2co.
_
dx dy
So that the £- component of the^curl of the linear velocity is twice
the angular velocity about the ^-axis. Further examples are presented
to us in the theory of fluid motion.
31. Lamellar Vectors. In finding the variation of the integral I
in the previous section, since the variations dx, dy, 82 are perfectly
arbitrary functions of s, if the integral is to be independent of the
path, dl must vanish, which can happen for all possible choices of
dx, dy, dz, only if
dZ dY_dX dZ_3Y dX _ n
~5i~'9il-~**Ji ~~ d*~ z>* 3y ~
that is if the curl of E vanishes everywhere. In case this condition
is satisfied, I depends only on the positions of the limiting points A
88 ni. GENERAL PRINCIPLES. WORK AND ENERGY.
and B, and not on the path of integration. Consequently, as stated
without proof in § 28, the conditions 38) are sufficient as well as
necessary.
If A is given, I is a point -function1) of its upper limit _B, let
us say (p. If B is displaced a distance s in a given direction to .Z?',
the change in the function cp is
B'
VB' — v dy 7
dx = x> dj==Y> Tz = z'
A vector whose components are thus derived from a single scalar
function qp is called the vector differential parameter of (p.
Accordingly the three equations of condition 38), equivalent to
curl R = 0, are simply the conditions that X, Y, Z may be represented
as the derivatives of a scalar point -function. In this case the
expression
Xdx + Ydy + Zdz = d^dx + d2ydy + d^dz = d mr -f + vmr - = 0.
But since A, p, v are perfectly arbitrary this is equivalent to the
three equations
Since the w's are independent of the time, we may differentiate
outside of the summation and write the above
d* d* d2
43) -^ 2rmrxr = 0, j^ Zrmryr = 0, -^ 2rmrgr = 0.
If we define the coordinates of a point x, y ~, ~z by the equations
and if we consider a mass m to consist of m particles of unit mass,
being the sum of the x- coordinates of the whole number of unit
particles divided by their number is the arithmetical mean of the
x- coordinates. If m is not an integer, by the method of limits we
extend the ifiotion of the mean in the usual manner. The point
x, if, ~s, the mean mass point thus defined is called the center of mass
of the system. (The common term center of gravity is poorly adapted
to express the idea here involved and had better be avoided. We
shall see in the chapter on Newtonian Attractions that bodies in
general do not possess centers of gravity.)
The equations 43) thus become
A A\ d*x ^ d*y „ d* z ^
44) 5F=°> d^ = 0> 5F-a
Therefore the center of mass of a system whose parts exert forces
upon each other depending only on their mutual distances moves
with constant velocity in a straight line. This is the Principle of
Conservation of Motion of the Center of Mass. It evidently applies
to the solar system. What the absolute velocity of the center of
mass of the solar system is or what its velocity with respect to the
so-called fixed stars we do not at present know.
32] MOTION OF CENTER OF MASS. 91
Returning to the equations 39), whether there is a force -function
or not, A, [i, v, being the same for each term of the summation,
may be taken out from under the summation sign and being arbitrary,
the equation 39) is equivalent to the three
or as before
46) d^Zrmr = ZrXr, ^Zrmr = ZrYr, ^Zrmr
that is: The center of mass of any system of the kind specified
moves as if all the forces applied to its various parts were applied
at the center of mass to a single particle whose mass is equal to
the mass of the whole system.
This principle of the motion of the center of mass reduces the
problem of the motion of the system to that of finding the motion
of a single particle together with that of the motion of the parts of
the system with respect to the center of mass.
A rigid body is a system of particles coming under the case
here treated, since the only constraints are such as render all the
mutual distances of individual points constant. Therefore the only
new principles required in order to treat the motion of a rigid body
are such as determine its motion relatively to its center of mass.
If the center of mass is to remain at rest or move uniformly,
we must have
47) 2;rxr = o, .zrrr = o, zrzr = o.
This will always be the case as shown above for mutually attracting
particles, since to. every action there is an equal and opposite
reaction. The three equations 47) furnish three necessary con-
ditions for the equilibrium of a rigid body.
If we introduce the relative coordinates of the particles with
respect to the center of mass into the expression for kinetic energy
it assumes a remarkable form. Let us put
xr = x + |r, yr = y + ijrf sr = ~8 4- £0
then
dxr dx di-r
~dt"'=~di^ ~dt'
dt ~ dt dt
zr dz d£
dt = = ~dt ~^~ ~dt
92 HI. GENERAL PRINCIPLES. WORK AND ENERGY.
48) r=
9— ^T-L 9^ ^!4.0^f ^
dt d< + d< d* + dt dt
r\* /Jflr\» / -~i -^>
- i[((f )'+(§)
+
Now in the last three terms we may write
if J is the a;- coordinate of the center of mass in the |, 17, g system.
But since the center of mass is the origin of the relative coordinates
%,ij,£, this is equal to zero. Similarly for the terms in rjr and £r.
Thus we have remaining if we write M for the mass of the whole
system,
The first term is the kinetic energy of a particle whose mass is equal
to the total mass of the system placed at the center of mass, while
the second is the relative kinetic energy of the system with respect
to the center of mass. Thus the absolute kinetic energy is always
greater than its relative kinetic energy with respect to the center of
mass (unless the center of mass be at rest). The center of mass is
the only point for which such a decomposition of the kinetic energy
is generally possible.
If the principle of the conservation of motion of the center of
mass holds we have
dx dy , dz _
~dt~a> ^di — °> 'di=*c>
32] RELATIVE KINETIC ENERGY. 93
and inserting these in the equation of energy for a conservative
system, T+W=h,
In this case accordingly the principle of conservation of energy holds
also for the relative kinetic energy, the constant h heing changed.
Inasmuch as we know of no absolutely fixed system of axes of
reference it is obvious that the kinetic energy of any system contains
an indeterminate part. But in virtue of the above principle if we
consider the center of mass of the solar system to be at rest all our
conclusions with regard to energy will hold good. The effect in
general of referring motions to systems of axes which are not at
rest will be dealt with in Chapter VII.
As a simple example of the above principle let us consider the
case of a rigid sphere or circular cylinder, with axis horizontal,
rolling without sliding down an inclined plane under the action of
gravity. If the distance that the center of the body has moved
parallel to the plane be s, the first part of T is -- -^M ^) • If the
angle that a plane through the horizontal axis parallel to the inclined
plane makes with the normal to the inclined plane be # (Fig. 23),
•j f\
the velocity of a particle with respect to the center is f-^t where r
is its distance from the horizontal axis. The relative kinetic energy
is thus
•7 f\
or since -j-> the angular velocity of rolling is the same for all terms
of the summation,
The factor 2rmrrl is called the moment of inertia of the system about
the horizontal axis through the center of mass and will be denoted
by K. Thus we have
If the rolling takes place without sliding we have the geometrical
condition of constraint,
- d& ds
where E is the radius of the rolling body.
94
HI. GENERAL PRINCIPLES. WORK AND ENERGY.
The loss of potential energy is M g times the vertical distance
f alien , ssina, where a is the angle of inclination of the plane to
the horizontal. Our equation thus becomes
l iTtf/ds\2l K {ds\z\
°4) T \M(di) + a* U) ) -
« - const.
If — = V when s = 0, determining the constant we have
Thus the motion is the same (cf. § 18) as that of a particle falling
freely with the acceleration diminished in the ratio
Fig. 23.
Thus by increasing Ky
which may be done
by symmetrically attach-
ing heavy masses to a bar
fastened to the cylinder
in such a way as not to
interfere with the rolling
of the cylinder (Fig. 23),
we may make the motion
as slow as we please and
thus study the laws of
constant acceleration.
33. Moment of Momentum. Under the supposition that the
equations of constraint were compatible with the displacement of
the system parallel to itself and that the force -function was thereby
unchanged we obtained the principle of the conservation of motion
of the center of mass. We will now suppose that the equations of
constraint are compatible with a rotation of the system about the
axis of X and that the force -function is thereby unaffected. This
will be the case in a rigid system or in a free system left to its
own internal forces (if conservative).
If we put
yr = rr COS C0r,
ob)
2r = rr sin «0
such a displacement is obtained by changing all the ror's by the
same amount do, leaving the r's unchanged. We have then
dxr = 0, dyr = —
^7 )
dzr = rr cos G)rdo =
32, 33] MOMENT OF MOMENTUM. 95
Inserting these values in d'Alembert's equation we obtain
58)
If U depends only on the mutual distances of the particles of
the system it is unchanged in the displacement, dU=0.
We then have
As was mentioned in § 11 the quantity within the parenthesis
is an exact derivative , so that
or differentiating outside of the sign of summation
Integrating we obtain
60) ^rmr \yr ™ — zr -—} = HX) an arbitrary constant.
The expression m(y~ — z^\=ymvz — 2mvy is the moment of
momentum [42), § 13] about the X-axis of the mass m, or it is the
•f Qf
product of twice the mass by the sectorial velocity -~^ (§ 8). The
theorem consequently states that the moment of momentum of the
whole system with respect to the X-axis is constant.
Under similar conditions for the other two axes we obtain
dx dz
dx
The vector H, whose components are Hx, Hy, Hz, is the resultant
moment of momentum of the whole system, and if the above equa-
tions 60) hold it is constant both in magnitude and direction. This
is the case for the solar system and we accordingly have an unvary-
ing direction in space characteristic of the system. This direction
was called by Laplace that of the Invariable Axis and the plane
through the sun perpendicular to it the Invariable Plane. It may
be defined as that plane for which the sum of the masses of each
particle multiplied by the projection of its sectorial velocity on that
plane is a maximum. Such a plane furnishes a natural plane of
coordinates for the solar system.
96 in. GENERAL PRINCIPLES. WORK AND ENERGY.
The principle expressed by equations 60) will be referred to as
the Principle of Conservation of Moment of Momentum. On account
of the connection with the sectorial velocity it has received the
shorter and more euphonious title of the Principle of Areas.
In case dU does not vanish, going back to equation 58), we may
divide out do and instead of 60) now obtain
dH
-jf = Zr(yrZr-8rYr),
dH
61) -jl = Zr(zrXr-XrZr),
dH
where Hx, Hy, Hz, have the same meaning as the left-hand members
of equation 60), but are not now constant. Stating in words: The
time derivative of the moment of momentum of any system with
respect to any point is equal to the resultant moment of all the
forces of the system about the same point.
The equations 46) and 61) furnish us the six equations of
motion of a rigid body. Geometrically, we may say that the radius
vector of the hodograph (§ 6) of the vector moment of momentum of a
system is parallel to the resultant moment of the forces acting on
tjie system at each instant of time, this statement being the com-
plement to the statement that the radius vector of the hodograph of
the velocity of the center of mass is parallel to the resultant of the
forces acting on the system.
The three principles which we have now treated, the Principle
of Energy, the Principle of Motion of the Center of Mass, and the
Principle of Moment of Momentum, in the cases of conservation, give
us the first integrals of the equations of motion, and suffice for the
treatment of all mechanical problems. In the next chapter we shall
deal with a principle which is more general than any of these in
that it enables us to deduce the equations of motion and thus
embraces a statement of all the laws of Dynamics.
IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 97
CHAPTER IV.
PRINCIPLE OF LEAST ACTION.
GENERALIZED EQUATIONS OF MOTION.
34. Hamilton's Principle. We shall now consider a principle
that differs from those of the last chapter in that it does not
immediately furnish us with an integral of the equations of motion.
On the other hand, like d'Alembert's principle it enables us to
embody the laws of motion in a simple mathematical expression
from which we can deduce the equations of motion, not only in the
simple form hitherto used employing rectangular coordinates, but also
in a form involving any coordinates whatsoever. This statement,
employing the language of the calculus of variations, permits us to
enunciate the principle in the convenient form that a certain integral
is a minimum. The so-called Principle of Least Action was first
propounded by Maupertuis1) on the basis of certain philosophical or
religious arguments, quite other than those upon which it is now based.
We shall first treat it in the form given by Hamilton. If in
d'Alembert's equation
we consider dx, dy, 8s arbitrary variations consistent with the equa-
tions of condition, we have
d*x ~ _ d (dx ,. \ dxdSx
~WOX ~di\dtox) ~~di~dt
d /dx ~ \ dx
= di\di°x)~~di
~dt
dt\dt
Treating each term in this manner, taking the sum, and removing
the sign of differentiation outside that of summation,
1) Mem. de 1'Acad. de Paris, 1740. Also: Des lois de mouvement et de
repos deduites d'un principe metaphysique , Berlin, Mem. de 1'Acad. 1745, p. 286.
WEBSTER, Dynamics. 7
98 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OE MOTION.
If there is a force -function U we have
Z(Xdx + Ydy + Zdz) = dU,
consequently the right-hand member of 1) is
dT+dU.
The left-hand member being an exact derivative we may inte-
grate with respect to t, between any two instants #0 and tl9
*
/
U) dt = d(T + U) at.
t0
If the positions are given for tQ and t±, that is if the variations
dx, dy, §z vanish for tQ and tlf then the integrated parts vanish, and
or
3) 8 C(T-W)dt = 0.
This is known as Hamilton's Principle.1) It may be stated by
saying ihat if the configuration of the system is given at two
instants tQ and t1} then the value of the time -integral of T -\- U is
stationary (that is less or greater) for the paths actually described in
the natural motion than in any other2) infinitely near motion having
the same terminal configurations.
Considering the signification of a definite integral as a mean3)
we may state equation 3) in words as follows: The time mean of
1) Hamilton. On a General Method in Dynamics. Phil. Trans. 1834.
2) It is understood that both the natural and the varied paths are smooth
curves, that is without sharp corners.
3) The arithmetical mean of a number of quantities is defined as their
sum divided by their number. A definite integral is defined as the limit of a
sum of a number of quantities as their number increases indefinitely. If we
divide the interval ab into n parts of length dg and if we denote by fg the
value of a function f(x) when x lies at some point within the interval §^ we
define
b
r
l
«/
f(x)dx as lim
34, 35] HAMILTON'S PRINCIPLE. 99
the difference of kinetic and potential energies is a minimum for the
actual path between given configurations as compared with infinitely
near paths which might be described (for instance under constraints)
in the same time between the same configurations; or more freely:
Nature tends to equalize the mean potential and kinetic energies
during a motion.
Hamilton's principle is broader than the principle of energy,
inasmuch as U may contain the time as well as the coordinates. It
is true even for non- conservative systems (where a force -function U
does not exist or where U contains the time), if we write instead
of dU,
We have then
4) {dT + Z(Xdx + Y8y + Zde)}dt = 0.
35. Principle of Least Action. It is to be noted that in
the statement of Hamilton's principle the infinitely near motion with
which the actual motion is compared is perfectly arbitrary (except
that it satisfies the equations of condition) ; so that to make the
system actually move according to the supposed varied motion might
require work to be done upon it by other forces. The paths described
by the various particles are not necessarily geometrically different
It is proved in the integral calculus that the manner of subdivision into the
intervals then dividing by (b — a) we have
that is the definite integral of a function in a given interval divided by the
magnitude of the interval represents the limit of the arithmetical mean of all
the values of the function taken at equidistant values of that variable throughout
the interval when the number of values taken is increased indefinitely. The
specification of the variable with respect to which the values are equally
distributed is of the first importance. For instance suppose that we change to
a new variable such that x = y(y], y = y-i(x) then
6 y
jf(x)dx = lf(x)q>
The integral may now be interpreted as the mean of the function f(x)
tan a = ~
Vx
and inserting the two values of t we get two possible elevations.
Thus we find that the aim is completely determined (though not
uniquely in this case) by the terminal positions and the velocity of
projection.
For the action we obtain
15)
A =
= m
- *)} dt
Using the values of Vz and t found above we obtain two values of
the action different for the two paths. Thus there are two possible
natural paths, differing from each other by finite distances, for only
one of which is the action least. Both however have the property
that between two points sufficiently near together the action is less
than for any infinitely near path.
In case the radical in 14) vanishes, that is
the two roots t2 are
equal and there is
only one course.
The terminal point
xlf #! then lies on
a parabola whose
vertex is vertically
above the point of
projection (Fig. 25).
It is easy to see
Fig 25 that this parabola is
the envelope of all
possible paths in this vertical plane starting from the same initial
point xQ1 £0 with the same velocity t'0. For it is the locus of the
35] KINETIC FOCI. 105
intersection of courses whose angles of elevation a differ infinitely
little. If the second point xlt y^ lie without this envelope it cannot
be reached under the given conditions. If upon it it can be reached
by one path, and if within it by two paths. In that case the course
that reaches xlf y± before touching the envelope has the less action.
A point at which two infinitely near courses from a given point
with equal energy intersect is called a kinetic focus of the starting
point, and if on any course the terminal configuration is reached
before the kinetic focus on that course, the action will be a minimum.
If the kinetic focus is first reached it will not.
Thus in the problem of motion on a sphere under no forces,
the point diametrically opposite the initial point is a kinetic focus.
Evidently a particle may reach the kinetic focus starting in any
direction from the original point, for all great circles through a
point intersect in its opposite point. The envelope of all the great
circles or courses from a point in this cases reduces to a point, which
is the kinetic focus.
For the treatment of the difficult subject of kinetic foci, which
belongs to the calculus of variations, the reader is referred to
Thomson and Tait, Principles of Natural Philosophy, § 358, and
Poincare, Les Methodes Nouvelles de la Mecanique Celeste, Tome III,
p. 261, also to Kneser^ Lehrbuch der Variationsrechnung.
From the principle of least action we may deduce the equations
of motion. Of course the principle was itself derived from these
equations, therefore, as is always the case, we obtain by mathematical
transformations no new facts. It is however instructive to see how
by assuming the principle of least action as a general principle we
may obtain the equations from it.
Let us put in equation 12)
fl<3 /7/V.2 I f]»3 l ,7*2
t*or — (*djr -J- ll'i/r \ W6r,
*~ - < -£ = y
dq ~ dq dq
giving
17)
If we put
since P involves all the coordinates and velocities xr, yr> zr, x'r, yl-, 0'n
106 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
19) iA
'io
Now the term
/;
2o
may be integrated by parts, giving
9
dP . /*. d /oP\ 7
o-l OXr / — I 0 Xr -y- ( 5—? ) »# •
0a?; r/ j r>?3 V'7^/
The terms in d^ and 60'r are to be treated in like manner. Since
the variations of the coordinates vanish at the limits the integrated
terms disappear, leaving
Now in virtue of 18) since N does not contain the coordinates,
dP
^ ~" Y M~Wx~r~ M
Also since M. does not contain #/.,
cP 1 T fM cN T fM.
^T == "« I/ -*r a~^ =1/^7
^^ 2 V N dxr V N
and consequently
dP d fdP\ -*/~N dW d
5~ -^-(^-i):=--l/irF^
^a; dcx' M ox d
M dxr\
^F-
N dqj
The equation of energy,
gives
or according to 18),
from which we get
35] EQUATIONS DEDUCED FROM LEAST ACTION. 107
Inserting this value of dq gives
dP _ d / dP \ _ -]/~N_ $W _ _^_
= _ ,/^ fdw
Accordingly we have
dW
In order that this may vanish for arbitrary variations, dxr, dyr, dzrj
the coefficient of each variation must vanish, so that we must have
dW
=0 or «-
c
d*zr dW d*zr dW
mr~w+~Wr= °> mr~dt^~~ ~fWf=*Zr\
which are the ordinary equations of motion for a free system.
The variations dxr, dyr, 8zr are arbitrary only if all the particles
are free. If there are constraints the variations must be compatible
with the equations of condition,
that is we must have the & linear relations between the 12, . . . ^, so that k of
the factors multiplying the variations vanish identically. Then the
108 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
coefficients of the remaining 3n — k arbitrary variations being put
equal to zero with these k give the differential equations
From the 3n equations 23) we may eliminate the It multipliers
Z1; A2, . . . A* and obtain 3w — k equations of motion , which is the
number of degrees of freedom of the system.
The equations 23) are known as Lagrange's differential equations
in the first form. They can evidently be deduced from equations 16)
of Chapter III by d'Alembert's principle , replacing Xr by
d*xr
Xr — ™r dtz > etc.
36. Generalized Coordinates. Lagrange's Equations. In
many investigations in dynamics where constraints are introduced,
instead of denoting the positions of particles by rectangular coordinates
(not all of which are independent) it is advantageous to specify the
positions by means of certain parameters whose number is just equal
to the number of degrees of freedom of the system, so that they
are all independent variables. For instance if a particle is constrained
to move on the surface of a sphere of radius I, we may specify its
position by giving its longitude cp and colatitude #, as in § 23.
These are two independent variables.
The potential energy depending only on position will be expressed
in terms of cp and #. The kinetic energy will depend upon the
expression for the length of the arc of the path in terms of cp and #.
Now we have, if / be the radius of the sphere,
Dividing by dt2 and writing #' = -j-> (p' = -~> we have
24) T = y m P (#' 2 + sin2 & y1 2).
The parameters # and cp are coordinates of the point, since when
they are known the position of the point is fully specified. Their
time -derivatives &',
&)> y = f* (ft > 0a)» * = /3 (0i, 0a)>
from these three equations we can eliminate the two parameters ql9 q2,
obtaining a single equation between x, y, s, the equation of the
surface. The parameters q± and q2 may be called the coordinates of
a point on the surface, for when they are given its position is
known. If q^ is constant and q2 is allowed to vary, the point x, yy s
describes a certain curve on the surface. This curve changes as we
change the constant value q±. In like manner putting q2 constant
we obtain a family of curves. The two families of curves,
ql = const, q2 = const,
may be called parametric or coordinate lines on the surface, any
point being determined by the intersection of two lines, for one of
which q^ has a given value, for the other, g2.
We may obtain the length of the infinitesimal arc of any curve
in terms of q1 and g2. We have
dx i , dx -.
25)
Squaring and adding,
26) ds* = dx* + df + dz* = Edq^ + ^Fdq, dq2
HO IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
Thus the square of the length of any infinitesimal arc is a homo-
geneous quadratic function of the differentials of the coordinates q1
and #2 , the coefficients E, F, Gr
being functions of the co-
ordinates q1} q2 themselves.
If the curve is one of the
lines q± = const , we have,
since dql = 0J
ds22=G-dq22,
if it is one of the curves
q% = const, we have
Considering any arc ds as the diagonal of an infinitesimal
parallelogram with sides ds^ and ds2 including an angle # (Fig. 26),
we have by trigonometry,
ds2 = ds±2 -f 2 dsj_ ds2 cos & -f- ds^.
Making use of the above values of ds± and ds2 and comparing with
the expression 26), we find
F
cos # = -=•
1/EG
If the coordinate lines cut each other everywhere at right angles we
shall have cos # = 0, F= 0, so that
28) ds2 = Edql2 + Gdq22.
The coordinates qi9 q2 are then said to be orthogonal curvilinear
coordinates. In the example above1) & and (p are orthogonal, the
lines of constant & and cp being parallels and meridians intersecting
at right angles and the product term in d&dcp therefore disappearing.
Employing the 'expression 26) for the length of the arc, dividing
by dt2 and writing
1) We have the equations of change of coordinates,
x = I sin # cos qp ,
y = I sin O1 sin qp ,
z = I cos -91,
from which
- = I cos & cos qp , -- = I cos & sin op , — = — I sin #,
<7aT C& Qv
— = _ ZsinO-sinqp, ~ = I sin -91 cos op , — = 0,
G
(Xrdxr + Yrdyr + Zrder)
is a homogeneous linear function in the dq's which we will write
40) d A = P4 dq, + P2 dq2 + • • • + Pmdqm.
By analogy with rectangular coordinates we shall call Pr xthe
generalized force -component corresponding to the coordinate qr and
velocity ql.
If the system is conservative, since
41) dW=-dA, Pr=-
and in any case
dXr + Y 8Vr 4 7 r
r Tr+Zr
r=l
We may now make use of Hamilton's Principle to deduce the
equations of motion in terms of the generalized coordinates q.
Performing the operation of variation upon the integral occurring
in Hamilton's Principle, both the g's and #"s being varied, we obtain
to
and since
dq
36, 37] LAGRANGE'S EQUATIONS.
we may integrate the second term by parts. Since the initial and
final configuration of the system is supposed given, the dq's vanish
at t = t0 and t = t^ so that the integrated part vanishes, and
Now if all the dq's are arbitrary, the integral vanishes only if
the coefficient of every dqs is equal to zero. Therefore we must have
45) d(T-W) _d_t<) (T-W}
dqs di ( dq'g -
or if we write L for the Lagrangian function T — W,
46) ±(**L\-?L.
' dt\d^)~dqs
Since the potential energy depends only on the coordinates
j- = 0, and we may write the equation 45)
There are m of these equations, one for each q. These are Lagrange's
equations of motion in generalized coordinates, generally referred to
by German writers as Lagrange's equations in the second form.
Their discovery constitutes one of the principal improvements in
dynamical methods and we shall refer to them simply as Lagrange's
equations.1)
If the system is not conservative, by § 34, 4) we must write
fl ft
48) l($¥4 8A)dt = l(dT
J J
from which we easily obtain 47), except that Ps is not now derived
from an energy function.
37. Lagrauge's Equations by direct Transformation.
Various Reactions. On account of the very great importance of
Lagrange's equations, it is advantageous to consider them carefully,
from as many points of view as possible. The deduction from
Hamilton's principle is one of the simplest, but does not perhaps
appeal as strongly to our physical sense as is desirable. Of course
as Hamilton's principle is completely equivalent to d'Alembert's, and
that to the equations of motion of Newton, we might have derived
1) Lagrange, Mecanique Analytique, Tom. I, p. 334.
116 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
the equations from either. This we will now do. It is important
every time that a new quantity appears in dynamics, to have a clear
conception of its physical nature. We should make free use of all
analogies that our science may offer us, and here geometry aids us
readily. The notion of the geometric product and the terminology
of multidimensional geometry here furnish us valuable aid. The
geometric product of two vectors in three dimensional space, defined
by their components X, Y, Z, X', Y', Z',
xx' + rr + zz',
is a scalar quantity, symmetrical with respect to both vectors, such
that the geometric product of the resultants of two sets of com-
ponents is the arithmetical sum of the products of all the pairs of
corresponding components. If one of the vectors is an infinitesimal
displacement dx, dy, dz, the geometric product is
Xdx + Ydy + Zds,
and the multiplier of the change dx is called the component of the
vector in the direction of the coordinate x. In like manner
let us speak of a quantity defined by components P1; P2, . . . Pm as
a vector in m- dimensional space. The geometric product of two
such, of which the second is an infinitesimal displacement compatible
with the constraints, and defined by the quantities dqi} dq%, . . . dqm,
may be, by analogy, defined as
Pldql -f P2 dq2 -\ h Pmdqm.
If now the vector P1? . . . Pm is equivalent to the system of vectors
Xr, Yr, Zr, we have equations 39), 40), 42), and the latter,
P,=
serves to define the component of the vector -system with reference
to the coordinate qs. Thus we have spoken of Ps as the force-
component of the system for the coordinate qs. It is to be observed
that we do not insist here on the idea of direction, and that our
terminology is merely a convenient mode of speaking. Nevertheless,
the notion of work gives a means of realizing by the senses the
meaning of our term component, for, if we move the system in such
a way that all the g's except one qs are unchanged the work done
in a change of the coordinate dqs will be P,^,.1)
Let us now find the component of our velocity -system according
to our generalized coordinates. We have, according to our equation
1) For a further elucidation of the nature of the geometric product, in
connection with multidimensional geometry, see Note III.
37] GENERALIZED VECTOR -COMPONENT. 117
of definition 42), for the component of the velocity of the rih particle
according to qs,
49) *£+*£+«£
Now we have by 32), dividing by dt,
-•-//*
The derivatives -~ contain only the coordinates q, not the velocities q',
which we see enter linearly, accordingly
Making use of this relation, the expression 49) becomes
Thus we find that the component of the velocity of any particle
according to the coordinate qs is equal to one -half the rate of change
of the square of its velocity as we change the velocity q'^1). This
result is not of itself of great physical importance, but leads us to
one that is. Inasmuch as the momentum is the important dynamical
quantity, multiplying by the mass of the particle we find
m^f! + **;fj + m,*;g - ~
or the component of the momentum of a particle according to any
coordinate is the rate of change of its kinetic energy as we change
the corresponding velocity. Summing for the whole system,
. 52)
that is, the component of the momentum of a system according to
any generalized coordinate qs is the rate of change of kinetic energy
1) It is to be observed that this "component" is not what we have called
the velocity q'g.
118 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
with respect to the corresponding velocity. The equation 38) now
says that the kinetic energy is one -half the geometric product of the
velocity and momentum systems. Thus we have perfect analogy with
the last two equations of § 27.
We shall hereafter denote the momentum belonging to qs by ps
and effecting the differentiation of 35) we have
O Q
or every generalized momentum -component is a linear function of
the velocities, the coefficients being the inertia -coefficients Qrs.
Let us now find the component of the effective forces according
to qs, the effective forces being defined by the system of products,
for each particle, of mass by acceleration,
dx[ dyl del
We have
to transform which we make use not only of 51), but of a relation
obtained as follows. Differentiating 50) by qs)
h
Using these results in 54), we obtain for the right-hand member,
- E «;) - *%] - iA k |*
and with similar results for y and 8j summing for all the particles,
we have for the component of the effective forces of the system,
d
dt\dq'
Putting the effective force equal to the applied force we have
Lagrange's equation 47) by direct transformation. The equation of
d'Alembert's principle thus becomes in generalized coordinates
56)
s = l
37] . EFFECTIVE FORCE COMPONENTS. 119
If we had begun with d'Alembert's principle we should evidently
have gone through precisely the same process that we have here
followed, and assuming all the displacements dx, 8y, dz to he virtual,
all the dg's would have been independent, so that from the trans-
formed equation 56) would have followed the individual equations 47).
This was in fact the mode of deduction followed by Lagrange.
We have a noteworthy difference between generalized and
rectangular coordinates, in that the effective force -component is not
dp
generally equal to the time -derivative of the momentum -~> but
dt
dT
contains in addition the term — «— • This we may accordingly call
the non-momental part of the effective force. Thus in general, even
though the momentum ps is unchanging, a force Ps must be impressed
dT
in order to balance the kinetic reaction -~— - As an example, let us
take the case of polar coordinates in a plane. We then have for a
single particle, for the coordinates qlf qz the distance r from the
origin, and the angle q> subtended by the radius vector and a fixed
radius. The kinetic energy is
from which we have the momenta,
dT
=
dT ,
Thus if the momentum pr is constant, which is the case when the
radial velocity r' is constant, we still have to impress a radial com-
ponent of force
The kinetic reaction —Pr = mrcp'2 is called the centrifugal force, a
name to which it is as much entitled as any sort of reaction is to
the term force.
By analogy we might in general call the non-momental parts of
the reversed effective forces or forces of inertia the centrifugal forces
of the system. These non-momental parts may be absent for some
coordinates. For instance in the present example (p does not appear
in the kinetic energy, but only its velocity (p'. We have then
£\ /j-j
- = 0. so that force need be impressed to change
1\ is a linear function
of the generalized accelerations $ '. Here again our generalized
coordinates differ from rectangular, in that there is a part of the
momental force which is independent of the accelerations #", but
which is a homogeneous quadratic function of the velocities,
r=m t=m o
58)
Consequently if at any instant of the motion we can change the signs
of all the velocities, and at the same time of all the accelerations,
the accelerational part of the momental force F^ will change its
sign, while the non- accelerational part F,W will be unchanged. We
may thus experimentally discriminate between the two.
Effecting the differentiation in the case of the non-momental
force, we find
which is also a homogeneous quadratic function of the velocities,
and thus possesses similar properties to Ff?\ Thus it is difficult to
discriminate experimentally between these two, unless we have some
experimental means of recognizing when the momentum ps remains
constant. In the simple example which we have used above, since
the non -accelerational part of the momental force belonging to r
disappears, while the centrifugal or non-momental does not, while for cp
37] VARIOUS TYPES OF REACTION. 121
although the non-momental part F^ = ^— disappears, we have the
non-accelerational part F^ = 2mr - r' y> . Experimentally this means
that, if a particle move with constant radial and angular velocities,
we shall have to apply to it not only a radial force F^ = — mry'2
to balance the centrifugal force, hut also a turning force 2mr-r'(p'.
This may he done by means of a varying constraint, say by making
a particle move upon a rod turning with angular velocity qp'.1) The
particle will then react upon the rod, to which the turning moment
2mr - r' cp' must be applied, for if it were not applied, owing to the
conservation of angular momentum, as the particle got farther from
the center its angular velocity would be less. To keep it constant
the particle must be pushed around.
We have now carefully analysed the effective forces, when
expressed in terms of our generalized coordinates. It is to be care-
fully borne in mind that all these parts come from real accelerations
impressed on the particles of the system, although the accelerations
of the generalized coordinates may disappear. This will depend on
our choice of such coordinates. The analysis that we have made is
however by no means devoid of physical significance, as we can not
usually observe all the bodies with which we have to do so as to
find their real motions and determine their accelerations, but are
obliged to become acquainted with them in a more or less round-
about way, through the reactions that they present to various
operations upon them. From this point of view it is of interest to
catalogue the various reactions that we meet in dynamics. In our
equation of d'Alembert's principle 56), we have called the P's which
are 'equated to the effective forces, the impressed forces, or forces
of the system. If the system is conservative, the forces of the
system are derivable from a potential energy, as we have assumed
in 47), while if not, part of the forces may still be derived from
such a function. It will be useful to consider not the forces of the
system, but the forces which must be impressed from outside in
order to counterbalance all the reactions of the system. In other words,
if we write Fs^ for the non- conservative part not yet dealt with,
60) Fs =
JFS is the force necessary to be impressed on the system from outside
under any circumstances whatever, or — Fs is the reaction of the
system, exerted through the coordinate qs.
1) The centrifugal force may be balanced by a spring.
122 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
If the system is left to itself, uninfluenced by other systems,
then every F9 is zero, and we have equation 47) with
If two systems are coupled together, so that any change of the
coordinate qs is accompanied by an equal change of the corresponding
coordinate of another system, then the Fjs of the two systems are
equal and opposite, which is the law of action and reaction. Accord-
ing to what happens to the system, the effect of Fs is of different
kinds. For instance, if the system is at rest, or moves very slowly,
all the jF/r) terms vanish except the last, and we have the static
reaction
3W ^
The work that is then done by the external forces,
is stored up as potential energy in the system. If there is no
possibility of statical storage, and if there is no non- conservative
reaction, we have only the kinetic reactions already dealt with.
As a simple example of what is meant, suppose the system to
consist of a mass attached to a spring tending to draw it to the
right. If the mass is at rest, it must be held by a force applied
from outside, to keep the spring stretched, and the static reaction
of the spring Ps is toward the right. If the mass is let go, it
begins to move toward the right, and the kinetic accelerational
reaction is toward the left, balancing the static reaction, or internal
impressed force of the system, according to d'Alembert's principle.
If there is no inertia, so that the effective forces vanish, and no
storage, the work done upon the system is not stored, but is said to be
dissipated. The reaction — F^ does not, in the cases that exist , in
nature, appear except when there is motion, that is, the reaction
- jpy4) is a kinetic reaction, though not due to inertia. This work
dissipated,
is always positive, in other words, non -conservative reactions are
always such as to oppose the motion. A case of frequent occurrence
is that where there are non -conservative forces proportional to the
first powers of the velocities q1, so that any F3W = KsqJ. We may
then form a function F which is, like T, a homogeneous quadratic
function of the velocities,
37]
CLASSIFICATION OF REACTIONS.
123
and since in this case the work dissipated in unit time is
F represents one -half the time rate of loss, or dissipation of energy.
F is called the Dissipation Function, or the Dissipativity.1) It was
introduced by Lord Rayleigh, and is of use in the theory of motions
of viscous media, and in the dynamical treatment of electric currents.
Beside this case we have dissipative forces not capable of representa-
tion o/f by a dissipation function.
We will now place our various reactions in a table showing
their grouping in various classes and sub -classes.
Positional
Reactions
Inertial
Motional
or Kinetic
,T , ,( Accelerational
Momenta!]
Non - accelerational F W
Non- momental
or Centrifugal
JT (8)
Non- conservative
Having Dissipation -function
Others
The advantage of this complete classification is as follows.
Suppose that a certain system or apparatus is presented to us for
dynamical examination. Its parts are concealed from o%r view by
coverings or cases, but at certain points there protrude handles, cranks,
or other driving points, upon which we may operate v and which will
exert certain reactions. All that we can learn of the system will
become known to us by a study of the reactions. Maxwell 2) compares
such a system to a set of bell -ropes hanging from holes in a roof,
which are to be pulled by a number of bell ringers. If when one
rope is pulled none of the others are affected, we conclude that that
rope has no connection with the others. If however, when one rope
is pulled, a number of others are set in motion, we conclude that
there is some sort of connection between the corresponding bells.
What the connection is we can find out by studying the motions.
In general, if when we move one driving point, and let it go, it
remains where we put it, we conclude that it is not attached to
anything, but is a mere blind member. If when we push it, it
1) A case of perhaps equal importance is one in which the dissipation
function contains the squares of differences of the velocities.
2) Maxwell, Scientific Papers, Vol. II, p. 783.
124 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
returns to its former position, we infer that it is connected with
something of the nature of a spring, and that the system can store
potential energy. If when we push it it keeps on going after we
release it, we conclude that it is connected with a system possessing
inertia, and capahle of storing kinetic energy. If its motion dies
away, we conclude that there is dissipation, and so on. By experi-
menting in turn, or simultaneously, on all the driving points, we
may conclude how many degrees of freedom the system has, how
the inertia is distributed, and how the parts of the system are
connected. The means of doing this will we discussed later, and we
shall find that in this manner we may learn much of a system, but
that our knowledge will not always be complete. This is the nature
of the process by which the physicist proceeds in the attempt to
explain recondite phenomena, such as those of heat or electricity,
by reducing them to the simpler phenomena of motion. The parts
of the systems, be they made of molecules of matter, or of the
ether, are concealed from him, but he may operate upon them in
certain experimental ways, and draw definite conclusions from the
results. One of the greatest triumphs of this method was Maxwell's
dynamical theory of electricity.
Impulsive forces are dealt with by Lagrange's equations in the
usual manner. Integrating equations 47) with respect to the time
throughout a vanishing interval t± — tQ, since the velocities are finite,
the non-momental forces — -«— are by 58) finite, so that the integral
of the second term vanishes, and we have
=(li-< p
ti — toj
P.dt.
Thus the momentum generated measures the impulse, as in the case
of rectangular coordinates, § 27.
As a further example of the use of Lagrange's equations let us
take the problem of the spherical pendulum, which we used to
introduce the subject. We had
24)
W =. — mgl cos &.
We have for the momenta p$. and p<
62)
cT
( uMVVfcfcSITY
Of
37,38] ENERGY INTEGRAL 'CGE'S EQUATION. 125
•
and our differential equations are
d , 72Q,N dT d
di(ml®)-d& = ~~
*(mvwt*.9')-.
dt^ ^ ^ Off dtp
Now since m and I are constant the equation for # becomes
64) ^ — sin#cos# • qp'2== -- ~- sin#,
in which the centrifugal force -component according to & is
The equation for qp (cp has no centrifugal part),
65) |^(72sin2# V) = 0
may at once be integrated, giving
66)
which is the integral equation 50), § 21.
Substituting in 64) the value of qp' derived from the integral
equation 66), we obtain the differential equation for #, which is the
same as the derivative of equation 51), § 21. The remainder of the
solution is accordingly the same as in § 21.
38. Equation of Activity. Integral of Energy. Let us
multiply each of Lagrange's equations by the corresponding velocity ql,
and add the results for all values of r, obtaining
The expression on the right, otherwise written
^^ d qr d A.
^Jrl>r~d^==~dt'
represents the time -rate at which the applied forces do work on the
system. The equation 67) is accordingly the equation of activity,
§ 27, 20), in generalized coordinates.
By means of the property of T expressed in equation 38), § 36,
we may transform the left-hand side of the equation, for, since T
depends upon both the g's and qns, both of which in an a'ctual
motion depend upon t, differentiating totally,
dT x^i/dTdtf dT d
126 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
•
Now differentiating 38) totally
dT dq'r
Subtracting equation 68) from 69) the terms ~ — -f -j— cancel and we
hav eft
dT „ f f d SdT\ dT dqr
But this exactly the left-hand member of the equation of activity 67).
Thus if the system is conservative, since
d^ - _ dw *•£ - dw
dt dt dt dt
so that the equation of conservation of energy is always an integral
of Lagrange's equations.
39. Hamilton's Canonical Equations. Although the equa-
tions of Lagrange are by all odds those most frequently used in
dynamical problems, yet in many theoretical investigations a trans-
formation introduced by Hamilton is of importance.
The kinetic energy being a quadratic form in the velocities qf
[equation 35)], the momenta pr being the derivatives of T by the
q"s are, as we have seen, linear forms in the q^s.
dT
53)
n n* _L n /•»' _i_ _i_ n ^
Pm = Q- / = = Vml #1 T Vm2^2 ~r ' ' ' ~T ^mrnqm-
OCLm
These linear equations may be solved for the g^'s, obtaining any
as a linear function of the jpr's, say,
n i \ . l ~D ., [ ~D ^ i i ~D „
the J?'s being minors of the determinant,
I Qml) Qm2> • - • Qmm i
divided by D itself.
The R's accordingly , like the §'s, are functions of only the
coordinates q. Maxwell calls them coefficients of mobility. The
solution of the equations assumes that the determinant D does not
vanish. This is always the case, being one of the conditions that T
is an essentially positive function.
39] HAMILTON'S EQUATIONS. 127
Let us now introduce into T the variables p in place of the
variables q', so that T is expressed as a function of all the 's
and j?'s. Since
38) **
inserting the values of q', in terms of the p's gives
72) T
that is T is now expressed as a quadratic form in the ^s. We will
distinguish T when expressed in terms of the ^'s by the suffix^, Tp.
We now have by Euler's Theorem,
dT
Since Tp is identically equal to T, comparing with equations 38),
above we have by symmetry,
thus the q"8 are linear forms in the p's given by 71). The two
identically equal functions, T, Tp, having the properties
dT dT
75) Pr,
are said to be reciprocal functions.1)
The expressions for the forces and potential energy are left
unaltered. Let us now make use of Hamilton's principle with this
choice of variables. Before performing the variation it will be
advantageous to introduce in the integral to be varied instead of the
Lagrangian function, L = T— W} the Hamiltonian function, H= T+W,
by means of the relation
76) L = 2T-H.
T and H are both to be expressed as functions of the variables q
and p, both of which depend upon the time t in a manner to be
found by integrating the differential equations of motion.
Hamilton's principle then takes the form
77) dl(2T -H)dt = d I Zr(prql - H)dt
J J
t0 to
tl
C^ri / f dH dH \
= I ^r I qr dpr -(- prdcir — ^ — $pr — ^ — o qr I (*^.
/ ^mJ > ^JP r ^ ^r '
#0
1) Webster, Theory of Electricity and Magnetism, § 63, 64.
128 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
The term prdq'r = Pr^. Sqr being integrated by parts and the dq's
put equal to 0 at the limits, we have
78)
Now since W does not depend upon the momentum prt
therefore the coefficients of the dp's all vanish. If the dq's are all
arbitrary, their coefficients must accordingly vanish so that we have
the first equation being the equation of motion, the second defining
q'r = —j-^- These equations 78) were introduced by Hamilton and on
account of their peculiarly simple and symmetrical form they are
often referred to as the canonical equations of dynamics. In practical
problems they are generally not more convenient than Lagrange's
equations.
We may recapitulate Hamilton's method as follows:
Form the Hamiltonian function H, representing the total energy
of the system as a function of the 2m independent variables q and p,
the coordinates and momenta. Then the time derivative of any co-
ordinate q is equal to the partial derivative of H with respect to
the corresponding momentum p> while the time derivative of any
momentum is equal to minus the partial derivative of H with respect
to the corresponding coordinate. A direct deduction of the equations
of Hamilton without the use of Hamilton's Principle will be found
in the author's Theory of Electricity and Magnetism § 64.
The equation of activity is most simply deduced from Hamilton's
equations, for by cross multiplication of equations 78), after trans-
posing and summing for all the coordinates we get
791 ^ SHdp.
r + -~ =
But this is equal to the total derivative of If by t,
dJS_
dt ~~ u>
which being integrated gives
H = h,
a constant. But since H = T + W, this is the equation of energy.
••
39, 39 a] VARYING CONSTRAINT. 129
If the system is not conservative ; there may be still some forces
which are derivable from a potential energy function. In that case
the Hamiltonian function is to be formed with that energy, but we
must add to the right of equation 78 a) the non- conservative force
- Fr^\ Thus our equations become
80) fe. - FV ^-a%
dt " " dqr *r > dt~frir'
The equation of activity then becomes
0 H dq dH dp. dqr
~ + ~ ~
or
dt
if there is a dissipation function.
iX
, 39 a. Varying Constraint. It may happen that the equations
of constraint contain the time explicitly, that is
xi> yu *!>••• xn> y*, **) = o,
9>2 & Xl9 Vl9 *1?V" **9-y«9 *n) = 0,
82) . . . ..........
Such a case is that of a particle constrained to move on a surface
which is itself in motion, say a sphere whose center moves with a
prescribed motion. The constraint is then said to be variable, and
the work done by the constraint no longer vanishes, for the surface
has generally a normal component in its motion, which causes the
reaction to do work. The variability of the constraint has an
important effect on the equations of motion. We can then no longer
determine the position of the system by means of a set of in-
dependent parameters, but must give not only their values, but also
the time. We rnay put
83)
from which, by the elimination of the g's, we may obtain equa-
tions 82).
1) cf. § 37, 60).
, Dynamics.
130 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
Differentiating now totally, we have
dy "by dy
dyr = -gfdt + ^dq,
or on dividing through by dt,
i dxr
= ~dt
3yr
85) y;=1nr
We have now in each x',y',z', beside the linear function of q[, q'2, ... q^,
a term independent of the qtJs, but which may be expressed
in terms of the coordinates q and t. On squaring there are accord-
ingly not only quadratic terms in the g"s, but also terms of the
first and zero orders. On forming the kinetic energy
s—1
86) T = |VV
we accordingly find that instead of being, as before, a homogeneous
function of the q"s, it contains not only quadratic terms, but also
terms linear in and others independent of the #"s. The effect of
these linear terms in the kinetic energy, whatever be their origin,
will be discussed in § 50.
40. Hamilton's Principle the most general dynamical
principle. We have seen in this chapter how by means of
Hamilton's Principle we may deduce the general equations of motion,
and from these the principle of Conservation of Energy. As
40,41] HAMILTON'S PRINCIPLE GENERAL. 131
Hamilton's Principle holds whether the system is conservative or
not, it is more general than the principle of Conservation of Energy,
which it includes. The principle of energy is not sufficient to
deduce the equations of motion. If we know the Lagrangian func-
tion we can at once form the equations of motion by Hamilton's
Principle, and without forming them we may find the energy. For
we have
L = T - W,
E=T+W,
Accordingly
87) E-2T-L
so that the energy is given in terms of L and its partial derivatives.
If on the other hand the energy E is given as a function of the co-
ordinates and velocities, the Lagrangian function must be found by
integrating the partial differential equation 87), the integration
involving an arbitrary function. In fact if F be a homogeneous
linear function of the velocities, the equation 87) will be satisfied
not only by L but also by L +• F. For, F being homogeneous, of
degree one,
, cF
Consequently a knowledge of the energy is not sufficient to find
the motion, while a knowledge of the Lagrangian function is. The
attempt has been made by certain writers to found the whole of
physics upon the principle of energy. The fact that the principle
of energy is but one integral of the differential equations, and is
not sufficient to deduce them, should be sufficient to show the
futility of this attempt. It is the infinite order of variability of the
motion involved in the variations occurring in Hamilton's Principle
that makes it embrace what the Principle of Energy does not.
41. Principle of Varying Action. We shall now deal with
a principle, likewise due to Hamilton, somewhat broader than that
which we have hitherto called Hamilton's Principle or Principle of
Least Action, and furnishing a means of integrating the equations
of motion. In the principle of least action a certain integral, belong-
ing to a motion naturally described by a system under the action
of certain forces according to the differential equations of motion,
has been compared with the value of the same integral for a slightly
different motion between the same terminal configurations, but not
a natural motion and therefore violating the equations of motion.
Under these circumstances the principle states that the integral is
9*
132 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
less for the natural motion than for the other. The new principle,
on the other hand, compares the integrals always taken for a natural
motion satisfying the differential equations , but the terminal con-
figurations are varied from one motion to another. The principle is
therefore known as the Principle of Varying Action.
In the process of § 34 equation 2) we cannot now put the
integrated part equal to zero,, but instead of 2) we shall have
*
The integrated part, which is the sum of the geometric products of
the momenta and the variations of the corresponding positions at
the end of the motion minus the corresponding sum at the begin-
ning, may now be transformed into generalized coordinates. The
integral
S=f(T-W)dt,
where T and W are expressed as functions of the time, appropriate
to any given motion (whether natural or not) depends upon the
terminal configurations, and is called by Hamilton the Principal
Function. The terminal configurations being given we had dS = 0.
Let us now find an expression for dS in generalized coordinates
corresponding to the expression above in rectangular coordinates.
Proceeding as in § 3|^ equation 43) we obtain
89 ^.'ll
Since the various motions are all natural ones satisfying the differ-
ential equations of motion, the factor of every dq in the. integrand
vanishes, so that the integral vanishes of itself, and dS is accord-
ingly expressed as a linear function of the variations of the initial
and terminal coordinates. Since W is independent of the 0"s and
cT
fi^i =pr, making use of the affixes 0 and 1 for the limits tQ and tv
we may write
90) 98 = Sr&1t£-Zr&a£,
an equation which could have been obtained from the considerations
regarding geometric products at the beginning of § 37. This
41] VARYING ACTION. 133
expression for the variation of S is of great importance , for by
means of it we can obtain a method of integrating the equations of
motion , and obtaining the coordinates q and momenta p at any
time t±. As we are now to consider the upper limit ^ as variable
it will be convenient to drop the subscript 1.
Suppose we have integrated the differential equations of motion
completely so as to obtain every coordinate as a function of the
time t, involving 2m arbitrary constants, cly C2, . . . c2m? the number
necessarily introduced in integrating the m Lagrangian equations of
the second order or the 2m Hamiltonian equations of the first order.
Let the integrals be
Differentiating these by t we obtain
from which by equation 53) we may .find the ^>'s as functions of t,
\jfjj pr — (pr \tj C^j C% y • . • C2m)'
These equations with 91) constitute 2m integral equations of the
system. ,
Inserting the particular value £0 in our integral equations we have
91') (fr = fr (t , C , C , .
93') VQ. = cp (t c c
We accordingly have the 4^m + 1 variables,
connected by 2m integral equations. We may thus choose any
2m + 1 of them as variables in terms of which to express the
remaining 2m.
For instance in the problem of shooting at a target § 35 we
saw that the motion was completely determined by the coordinates
of the initial and final positions and the initial velocity. The latter
determined the time of transit £, so that it together with the initial
coordinates, q^0, . . . 0m, and the final coordinates, qlf . . . qm, may be
taken as independent variables in terms of which everything may
be expressed.
134 IY. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
Thus the integral
94) S
is supposed to be expressed in terms of these 2m + 1 variables.
Now if the initial and final coordinates are varied without varying
the time of transit t — tQ (t the upper limit of the integral) we have
We have however proved that under these conditions we have
Since these expressions must be equal for arbitrary variations of
the #'s and #°'s we must have
96) M= M^ 3S =
We may now, if we please, regard the initial coordinates
#i°7 • • . qm, and the initial momenta, p^9 . . . p£y as 2m arbitrary
constants replacing the c±, C2, . . . c^m of equations 91) and 93). Then
the equations 97) will be the general integrals of the equations of
motion, for if the form of the function S is known in terms of
tj &, . • • qm, #1°, - . . qm, the equations 97) are m equations involving
qly . . . qm without their derivatives, which may be solved to obtain
the g's as functions of t and 2m arbitrary constants q^9 . . . g£> J^ . . .pH,
as in equations 91).
It has appeared as if in order to find S it were necessary to
integrate the equations of motion, so to obtain T — W as a function
of the time, which being integrated would give S. If this were so
the statement just made would be of little interest. But this is not
necessary, for Hamilton showed that the function S, which he called
the Principal Function, satisfies a certain partial differential equation,
a solution of which being obtained, the whole problem is solved.
The function S is a function of the variables g, the constants (f
and the time t, which thus occurs explicitly and implicitly. Differen-
tiating by t we have therefore
«9 %-%
.
Differentiating 94) by t, the upper limit, gives however
dJl=T-W.
at
41] PARTIAL DIFFERENTIAL EQUATION. 135
Equating the two values,
by 38).
Transposing and writing T + W = H,
The function .0", the sum of the energies, depends upon the co-
ordinates qr and the momenta, pr = ^ — If the force -function depends
upon the time H will also contain t explicitly. Thus we have the
partial differential equation
oo\ dS .
+
The equation is of the first order since only first derivatives of S
appear, and, from the way in which T contains the momenta [equa-
o ci
tion 72)], is of the second degree in the derivatives « — Since S
appears only through its derivatives an arbitrary constant may be
added to it.
Thus we have the theorem due to Hamilton: If qlf . . . qm, ex-
pressed as integrals of the differential equations in terms of t and
2m arbitrary constants q^, . . . q&, p^, . . .p£, are introduced into the
integral 94), and the result is expressed in terms of t, qlf . . . qm,
(Zi0, . • • #m, then 8 is a solution of the partial differential equation 99).
The converse of the proposition was proved by Jacobi, namely,
that if we take any solution of the equation 99) containing m arbi-
trary constants, q^, . . . q£ (other than the one which may always
be added), the equations 97) obtained by putting the derivatives
of S by the m arbitrary constants equal to other arbitrary constants,
p^j . . . pm wiU be integrals of the differential equations of motion.
For the proof of this the reader is referred to Jacobi, Vorlesungen
tiber Dynamik, XX.
Before giving examples of the utility of this method we shall
show that the arbitrary constants by which we differentiate need not
be the ^°'s? but may be any m constants appearing in the integral
equations.
Suppose that in equations 91) we vary m of the arbitrary
constants c1; . . . cm. We then have
136 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION,
and putting t = 0,
Then equation 90) becomes
100) 68= Zrpr $ P,r = 3j-f> P>r=Wr
and equation 105) then is
In the case of a single particle comparing equations 110) and
111) we have
dA f dA
w me ~w
In other words if the action A is expressed in terms of the co-
ordinates x, y, s, the momentum of a particle describing any path
138 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION.
under the action of the given forces with the constant energy h is
the vector differential parameter of the action A, and therefore, by
the properties of lamellar vectors (§ 31), the velocity of a particle
moving in this manner is normal to all the surfaces of constant
action, and is inversely proportional to the distance between two
infinitely near surfaces of constant action. Otherwise expressed, if
from all points of any surface particles be projected normally with
the same energy h, their paths will always be normal to a set of
surfaces, and the action from one surface to another will be the
same for all the particles. This theorem is due to Thomson and Tait.1)
Suppose first there are no forces acting, then equation 112)
becomes
which is satisfied by the linear function
115) A = ax + ~by + C89
if
116) a2 -f 62 + c2 = 2wft.
In virtue of this last equation only three of the constants
a, 6, c, li are arbitrary. Suppose we take a, 6, ft, then we have
117) A = ax + ~by-
Then equations 107) or 113) are
118) mx' = a, my' = 1, mz' =-\/2mJi - (a2
which are first integrals of the equations of motion, showing that
the motion of the point is uniform. Equations 108) and 109) are
dA as =
~
dA lz
119) ~n=y-
y%mh-(a*-\- &2)
3 J. m#
/ /
6 v/i
^^ l/^-mTj — r«2-l-?i^ °
The first two of these equations are the equations of the path,
showing it to be a straight line, while the last gives the time. By
means of it we may find 2 as a function of the time, and from the
first two x and y. Thus 119) are the integral equations of the
motion.
Corresponding to this solution, a and & being constants, the
surfaces of constant action are parallel planes. The path of any
1) Natural Philosophy, § 332.
41] SURFACES OF EQUAL ACTION. 139
particle projected normally to one of these planes with the energy
(kinetic) h is a straight line normal to these planes, and the velocity
is constant.
In order to find solutions suited to surfaces of equal action
having other forms, we should require to find other particular
solutions of equation 114), which would take us too far into the
subject of partial differential equations. Whatever the nature of the
surfaces, since the velocity in all the motions considered is constant,
the action is proportional to the distance traversed and consequently
if we measure off on the normals to a surface of constant action
equal distances, the locus of the points thus obtained will be another
surface of equal action, or all the surfaces of equal action are so-
called parallel surfaces.
Next, suppose we have a single particle of mass unity under
the action of gravity. Then
w=ge,
and our equation is
TT l[/dA\*. /dA\* , /d-4\2) 7
120) H = ~ + + + g, = ft.
We may find a solution
A = ax + by + ),
where the functions R and F contain only the variables indicated.
= =
dr~~dr' d» ~ d&'
Substituting in equation 135) we have
1q7v 1
Multiplying by r2 and transposing
1QQ\ X
138)
On one side of this equation we have functions of r alone, on the
other functions of # and
^= /,!y"2-2l+^ jT/tT-%
*s V r rz *s f sm2
dA _ r d^ y _ f
148) ^ ~ / ^T/2^8in4^-ysm2^ 2 V7 ~~ ^ ;
a^i /* dr
Th=h
V"-
If we put y = 0, necessitating according to the second equation
9 = 0, the first equation becomes
149) v ' ' ^'
the equation of the path, which, on performing the integration
indicated, takes the form obtained in § 20 equation 23).
144 V. OSCILLATIONS AND CYCLIC MOTIONS.
CHAPTEK V.
OSCILLATIONS AND CYCLIC MOTIONS.
42. Tautochrone for Gravity. A curve along which a
particle will descend under the action of gravity to a fixed point
from a variable point in the same time is called a tautochrone curve.
If the particle is dropped from rest we have the equation of energy
and the time of falling to the level z = 0 is
0
Let the length of the arc s measured from the fixed point be q>(0),
then
3) t--
o
If the curve is to be a tautochrone this must be independent of #0 or
Let us change the variable by putting 3 = 00u, then
i
o
or changing the variable back to z9
o
If this is to vanish for all values of the limit #0 the integrand
must vanish, or
4) 9/0) + W(*) = °>
which is the differential equation of the curve. Writing this
y"(«) = !_
g>'(«) 2/
42] CYCLOID AS TAUTOCHRONE. 145
we may integrate, obtaining
log y' (z) = — 2- log z + const.
Taking the antilogarithm,
r-N . f x C dS
5) ^0) = -= = r>
]/2 d#
since s =
L.
Calling its roots A1; 12 we have
28) ^ = -1
The general solution is obtained by multiplying the- particular solu-
tions e*1* and e^ by arbitrary constants and adding. Thus we obtain
29) s
We have to consider two cases,
I. K2 >
II. %2 <
In case I the radical is real, and since its absolute value is less
than ;c both Aj_ and ^ are negative and s eventually decreases as the
time goes on, vanishing when t = oo. We have
This vanishes when
or
Consequently if 5 and ^i are of opposite signs s will increase to a
maximum and then continually die away. If they are of the same
sign the motion dies away from the start. Both cases are shown in
Fig. 31, where t is the abscissa and s the coordinate.
In case II the radical is imaginary and both ^ and A2 are
complex. Then writing
150
V. OSCILLATIONS AND CYCLIC MOTIONS.
= + v l =
— v
and making use of the fundamental formula of imaginaries,
31) eivt = cos vt + i sin vt,
and the principle that both the real part and the coefficient of i in
the imaginary part of a solution are particular solutions, we obtain
the two particular solutions
eutcosvt and
Fig. 31.
We thus obtain the general solution
32) s = e>ut(Acosvt + Bsinvf)
(A and B being new arbitrary constants),
or as in § 19 equation 10),
33)
The trigonometric factor represents a simple harmonic oscillation,
which on account of the continually decreasing exponential factor
dies away as the time increases (Fig. 32). Such a motion is called a
damped oscillation, and 7, is a measure of the amount of damping.
The extreme elongation occurs when
34) % = a
that is when
35) tan
2 J. \ *
— X2't—CC] = -
/ 1/A.lt*-.it*
43]
LOGARITHMIC DECREMENT.
151
The smaller the damping ?c, the more nearly does the time of the
maximum coincide with that of the maximum of the cosine factor
in 33). In any case successive maxima follow each other at intervals
equal to the period of the oscillation ,
36) T = ^
At two successive maxima on the same side, s1 and S2, the cosine
term will have the same value , therefore the ratio of the elongations
will be that of the exponential factors , or
Fig. 32.
The logarithm of the ratio,
37) (5 = ^^
-1
is accordingly constant, and by means of observations on the loga-
rithmic decrement we may determine the damping. We see that the
decrement depends on and increases with the ratio of the square of
the coefficient of damping % to the coefficient of "stiffness" I?.
If there were no damping, B = 0, we should have for the period,
Introducing these values of T0 and d, we may write
38) T =
152 V. OSCILLATIONS AND CYCLIC MOTIONS.
so that if the damping is small, as is usually the case, it affects the
period only by small quantities of the second order.
As has been shown in § 38 we have here an instance of the
use of a dissipation function
and the energy is dissipated at a rate proportional to the exponen-
tial e~xt.
44. Forced Vibrations. Resonance. The motion considered
in the last section being that of a system left to itself is called a
free oscillation or vibration. We shall now consider a problem of a
different sort from. any yet treated and involving a force depending
upon the time, and thus introducing or withdrawing energy from
the system. Let us suppose a particle to be subject to the same
conditions as above, but in addition to be acted upon by an
extraneous force varying according to a harmonic function of the time,
40) F=Eco$pt,
so that the differential equation of motion is
We may find a particular solution by putting
s = acos(pt — a),
42) ds . f , d*s o / j \
-^= -apsin(pt-a), -^ = - aj»8cos (pt — a).
Substituting in the differential equation, we have
43) a (h2 —p*) cos (pt — a) — axp sin (pt — a) = Ecospt
= E {GOS a cos (pt — a) — sin a sin (pt — a)}.
This can be identically true for all values of t only if the coefficients
of the sine and cosine of the variable angle (pt — cc) are respectively
equal on both sides of the equation, accordingly we must have
axp =
a(h2-
from which eliminating first E and then a,
xp
46) tan« = ^,
from which we obtain the amplitude
46) a = - =£
42, 43] PHENOMENON OF RESONANCE. 153
Thus our solution is
v
47) s = - ^ cos (pt — a).
y(fc»_p»)* + xv
The motion represented by this solution is called the forced
vibration, for the system is forced to assume the same period as
that of the extraneous force F, namely -— > of frequency — ; while
the frequency of the free or natural vibration would be —
or without damping - —
The displacement is not in phase with the force, lagging behind
it by less than a quarter -period if tana is positive, that is, if h is
greater than p, in other words if the natural frequency is greater
than the forced. If on the contrary the natural frequency is less
than the forced, tana is negative, and since sin a is positive, the
displacement is between a quarter and a half -period behind the force.
If the frequencies of the forced and free vibrations coincide, tana
becomes infinite, the lag is a quarter period, so that the displacement
is a maximum when the force is zero and vice versa. Then 47) becomes
48) s = — sin Jit.
p*
and if the damping % is small, the amplitude is very large. This is
the case in the phenomenon of resonance, of great importance in
various parts of physics, including acoustics, electricity, and dispersion
in optics. The equation shows how a very small force may produce
a very large vibration if the period coincides nearly enough with
the natural one, and explains the danger to bridges from the accu-
mulated effect of the measured step of soldiers, the heavy rolling of
ships caused by waves of proper period, and kindred phenomena.
Although in the phenomenon of resonance the excursion and
consequently the kinetic energy becomes very large, it is of course
not to be supposed that this energy comes from nothing as has been
frequently contended by inventive charlatans proposing to obtain vast
stores of energy from sound vibrations.1)
If we form the equation of activity, by multiplying 41) by ^>
inx 'd(T-\-W) . (ds\* 1 d i/ds\2
-dt~ - + *(di) = *di[(di)-
-E*p
= (— cos a sin pt cospt + sin a coss
^2
1) Of these the United States has produced more than its share. The
ignorance of the above mentioned principle enabled John Keely to abstract in
the neighborhood of a million dollars from intelligent (!) American shareholders.
154 V. OSCILLATIONS AND CYCLIC MOTIONS.
we see that energy is being alternately introduced into and withdrawn
from the system by the extraneous force. On the average however,
as we find by integrating the trigonometric terms with respect to
the time, T
/*
smptcos£)tdt = 0,
—
P
/•
the time average of the activity depends upon the last term containing
sin a, and this is always positive, consequently the extraneous force is
on the whole continually doing work on the system, which is being
dissipated at the rate xl-^} • This work is a maximum when a = — -9
\dt/ 2
when the system is in complete resonance. Thus the mechanical
effects producible by resonance are shown to be commensurate with
the causes acting, and the impossibility of the common story of the
fiddler fiddling down a bridge is demonstrated.
The exactness of "tuning", or approach to exact coincidence of
period necessary for resonance is shown in Fig. 33, which is the
graph of the curve
where y = -=?- is the ratio of the actual amplitude of equation 46) to
T?
the steady statical displacement p produced by a constant force E
(that is when p = 0), x = j- is the ratio of the frequencies of forced
and free vibration, and «2 = ^-1) The curves are drawn for values
of the parameter a2 equal to -01, -05, -10, -15, -20. Thus the magnitude
of the resonance for any particular case can be seen by a glance at
the figure. The resonance is sharper the smaller a. The maximum
amplitude is not for perfect tuning, but for x = 1/1 — — - The value
of the maximum is nearly equal to —
If there is no friction , for p = h the vibration becomes infinite,
which means simply that in this case friction must be taken into
account. If there is no friction we have by 44),
sinc£ = 0, cos a = 1
1) This parameter a. is not the angle cc above.
44]
EFFECT OF TUNING.
155
and the displacement is in the same or opposite phase with the
force , according as h is greater than or less than p. In the latter
case the excursion is a maximum in one direction when the force is
exerting a maximum pull in the opposite direction. This need not
appear paradoxical, for consider the limiting case of a system with
very little stiffness in proportion to its inertia, that is li very small
and the natural period very great. Then the excursion is always
opposite in phase to the force on account of the inertia of the
system. In the opposite case of a system with very little inertia in
proportion to the stiffness, h is very large, and the excursion is in
phase with the force. In this case (that of complete agreement) we
have what is- called the equilibrium theory of
oscillation, the displacement being the same as
(~F1\
S = ph
except that the force and displacement are varying
together. Such a theory was given by Newton
for the tides, which consist of a forced vibration
of the water covering the earth under the periodic
force due to the moon's attraction. The more
accurate theory taking account of inertia was
given by Lagrange. The relation of the dyna-
mical to the equilibrium theory is shown in Fig. 33.
The two points of distinction between free
and forced oscillations then are, first, that the
free vibration has its period determined solely
by the nature of the system, while the forced
vibration
takes the
period of the
P. force, and
secondly,
that if there
is damping,
the free vibration dies away, while the forced vibration persists
unchanged.
The theory of the forced vibration which we have given does
not take account of the gradual production of the motion from a
state of rest, but refers only to the motion after the steady state
has been reached. We may now complete the treatment and take
account of the motion at the start. Our previous solution is merely
a particular solution. According to the theory of linear differential
equations in order to obtain the general solution we must add to the
particular solution just obtained the solution of the equation 41)
Fig. 33.
156 V. OSCILLATIONS AND CYCLIC MOTIONS.
when the second member is equal to zero, or in physical terms the
forced and free vibrations exist superimposed. Accordingly we have
If the system starts from rest we must determine A and /3 so that
when t = 0, s and -^- are equal to zero. These conditions will be
very nearly satisfied, if p and h are nearly equal and x small, by
( i ^
I ~~9y't (-\ / x2 Yl
52) s = a\cos (pt — a) — e cos ( J/ft2 — — • t — a \ r
The simultaneous existence of two harmonic vibrations of nearly
equal frequencies gives rise to the phenomenon known as beats.
Suppose
53) s = a cos (pt— a) + & cos {(p + Ap] t — /3),
where Ap is a small quantity, equal to 2 x times the difference of
frequencies. We may write the last term
& cos {(pt — a) + dp - t -f a — /3)
= 6 {cos (pt— a) cos (dp • t + a — /3)
- sin (pt — a) sin (dp -t-\- a — /3)},
so that
54) s = {a -f- & cos (Ap • t + a — /5)} cos (pt — a)
- & sin (dp - 1 -f a — /3) sin (^^ — a),
or if we write
a -f & cos (z/# • £ -f- a — /3) = A
- & sin (z/p • t -f a — /3) = 5,
55) 5 = Dcos(pt — a — f),
where
and
D =
Accordingly the compound vibration may be considered as a harmonic
motion of variable amplitude and phase, the amplitude varying from
a + ~b to a — &, with the period -^- and frequency -^- equal to the
difference of the frequencies of the two constituents. The phenomenon
of beats or interferences is represented graphically in Fig. 34.
44, 45]
PHENOMENON OF BEATS.
157
In the case of free and forced vibrations coexisting [equation 52)],
we have at the beginning beats which gradually die away owing to
the factor e 2 in the free vibration, leaving only the forced vibration.
Fig. 35.
Fig. 34.
This is shown in an interesting manner by a tuning fork electrically
excited by another fork not quite in unison with it, the phenomenon
of a single driven
fork apparently
producing beats
with itself being
very striking (Fig.
35). It will be
noticed that the first maximum is greater than the steady amplitude.
The greater part of this section and the preceding is taken
from Rayleigh's Theory of Sound.
45. General Theory of small Oscillations. Having now
set forth the general characteristics of vibrations excuted by systems
possessing one degree of freedom, we will now treat the problem of
the small vibrations of any system about a configuration of equili-
brium after the manner of Lagrange, who first investigated it.
Suppose a system is defined by n parameters qi9 q2, . . . qn- Its
potential energy will depend only on the coordinates q, and developing
by Taylor's Theorem,
56) W = TF
where the suffix zero denotes the value when all the g's are zero.
Suppose that this is a configuration of equilibrium, then W is a
. . . (dW\ -, m, TT;r TT;r
minimum or maximum and every (-~ — I equals zero. Inus W - rK0
begins with a quadratic function of the #'s. If the motion is small
enough we may neglect the terms of higher orders of small quantities.
Accordingly, neglecting the constant W0 (for the potential energy
always contains an arbitrary constant which does not afPect the
motion), we shall put W a homogeneous quadratic function of the #'s
with constant coefficients,
r — n s = n
57) W=
158 V. OSCILLATIONS AND CYCLIC MOTIONS.
If the equilibrium is stable the potential energy must be a minimum
so that the constants crs will be such that the quadratic function W
is positive for all possible values of the variables q.
The kinetic energy will be a quadratic function of the time
derivatives, q[, q'2, . . . qi,
58) T=
where the a's are functions of the coordinates q alone. We may
develop the functions ars in series, thus one term of the sum becomes
59) ctr.qlq,' =
*=1
and since the velocities q' are small at the same time as the co-
ordinates g, we may neglect all the terms within the braces except
that of lowest order aj,, therefore we may consider the a's as
constants. If we have besides the conservative forces of restitution,
arising from the potential energy W, non- conservative resistances
which are linear functions of the velocities, we may make use of a
dissipation function F, § 39, such that the dissipative force correspond-
ing to the coordinate qr will be - - ~— 7- We thus have the three
homogeneous quadratic functions with constant coefficients,
r—n « = !
60) * = -g
r=l * = :
Each of these has the property of being positive for all possible
values of the variables of which it is a function. The a's may be
called coefficients of inertia, the c's, coefficients of stiffness, and
the jc's, coefficients of viscosity or resistance. We may now form
Lagrange's equations for any coordinate qr.
dpr dT dW dF
^t~Wr~^= ~Wr~Wr
where
45]
SMALL OSCILLATIONS.
159
62)
dT
-—
H ----- h ar*qn
H h C™_by substitution of any one lr
jjn^J^e^ji^tip^^ ratios ^At : A2 : - • - : An. For each
value ofjlr we obtain a different set of ratios. We will distinguish
the values belonging to Ar by an upper affix r, so that Ars means
the coefficient of e*r* in the coordinate qs.
The theory of linear differential equations shows us that for the
general solution we must take the sum of the particular solutions
Arse rt for all the roots Ar, so that we obtain
, Dynamics. 11
152 V. OSCILLATIONS AND CYCLIC MOTIONS.
73)
It is to be noticed that the ratios of the As in any column
have been determined by the linear equations 64), so that there is
a factor which is still arbitrary for each column, that is to say,
2n in all. We may now replace the exponentials by trigonometric
terms. The appearance of the terms with conjugate imaginaries
Are1* + A[ert = 2e^ (ar cos vt - pr sin vt)
leads to the disappearance of imaginaries from the result. Changing
the notation we will accordingly write
= % e^ * cos Vt - + B e^ cos
74)
If these be substituted in the differential equations it will be
found that the J5's satisfy the same linear equations as the As.
Each column then contains an arbitrary constant as before, in the .B's
and a second arbitrary constant in the 8 belonging to the column.
We may therefore state the general result: - - The motion of any
system, possessing n degrees of freedom, slightly displaced from a
position of stable equilibrium may be described as follows: Each
coordinate performs the resultant of n damped harmonic oscillations
of different periods. The phase and damping factor of any simple
oscillation of a particular period are the same for all the coordinates.
The absolute value of the amplitude for any particular coordinate is
arbitrary, but the ratios of the amplitudes for a particular period for
the different coordinates are determined solely by the nature of the
system, that is, by its inertia, stiffness and resistance coefficients.
The 2n arbitrary constants determining the n amplitudes and phases
are found from the values of the n coordinates q and velocities q'
for a particular instant of time.
45] NORMAL COORDINATES. 163
We further notice that, since the different periods depending
upon v are derived from the roots of an algebraic equation, they
are not in general commensurable, so that the motion is not as a
whole generally periodic. For instance in the case of Lissajous's
curves described in § 19, unless the two periods are commensurable
the curve will never close. In the case, however, of the spherical
pendulum performing small oscillations the periods of the two co-
ordinates were equal, so that the path became a closed curve, an
ellipse.
There is one set of coordinates of peculiar importance. For
simplicity let us suppose there is no dissipation, F—Q. Let us
make a linear transformation with constant coefficients, putting
& = yn 9>i + fia 9>a H f- nn(1 = o , - 1, C , - 1, 0. , . . . = 0; » rows.
0 , 0 , - 1, C ,-!,...
166
V. OSCILLATIONS AND CYCLIC MOTIONS.
Expanding the determinant in terms of its first minors we have
This equation between three consecutive determinants of the same
form suggests a trigonometric relation, namely, making use of the
relation
sin (a -j- &) -f sin (a — 6) = 2 sin a cos Z>,
with 1} = ft, a = n&, we have
sin (n + 1) # -f- sin (n — 1) #• = 2 sin w# cos #.
Comparing this with the formula 89),
we see that they are identical if we put
C = 2 cos #, Dn = fc.sin (n -f 1) #,
where c is independent of n. To find it put n = 1,
sin 2#
A
2 cos
Accordingly
90)
Dn =
sin (w -f- 1) •fl-
am #•
If this is to vanish we must have
where Jc is any integer (not a multiple of n -f 1, to prevent sin # in
the denominator from vanishing). Introducing the values of # thus
found we obtain
91)
from which
92)
00
== 2
\
\
Fig. 37.
87)
~ ~
= 2 cos # = 2 cos
— COS
n+l
Letting & = 1, 2, 3, . . . n, we obtain n different
frequencies proportional to the abscissae of
points dividing a quadrant into (n + 1) equal
parts, Fig. 37. Giving ~k other values not
multiples of (n + 1), we shall merely repeat
these frequencies. There are accordingly n
different frequencies for the vibrations.
We may arrive at the same result by
noticing that the linear equations for the -4's,
CAr -
= 0,
46] EQUATION FOR PERIODS. 167
are satisfied by
As = P smsft,
where P is a constant, making use of the same trigonometric for-
mula as before. Accordingly let us substitute in the differential
equation
ma dly
84) - yr_t + ^ -^ + 2yr - yr+1 = 0
the solution
93) yr = Psinr-frcos^ — s).
Every term will contain the same cosine, so that dividing out we have
- sin(r- 1)# + 2 l -~ - sinr# - sin(r + l)fr = Or
which is an identity if
giving
o a & /^ _ \
v2 = — (1 — cos#),
ma v
as before, 92). The complete solution is then
s = n
94) yr =^PS sin ^ cos (vf« - «,),
S=:l
with the 2w arbitrary constants Pg, cc5 to be determined by the initial
displacements and velocities.
Consider the case first in order of simplicity, n equals 2. Then
->-\nr . * -I/IT
v = 2 I/ — sin — = I/ — )
. *• \ ma 6 r ma
95)
o -
2 I/ - - sm — =
ma 3
Thus the frequency of the higher pitched vibration is in the ratio
of ~/3 : 1 = 1.732 to that of the lower, — somewhat more than the
musical interval of a sixth. In this particular case it is easy to find
the normal coordinates. Writing
96) ±
± = 0, f/2 = ~~ 2/i ? an(i the ^wo beads swing in
opposite directions with a frequency }/3 times as great as before.
The middle point of the string is now at rest, or forms a node.
The general case above treated is very interesting when we pass
to the limit as the number of beads is increased, giving us the case
of a continuous string, of the greatest importance in the theory of
musical instruments.
Let us introduce in equation 94) the distance of the bead from
one end of the string,
rl .
x = r • a = — ;— - *
Accordingly 94) becomes
s—n
100) y (x) = x< P* sin — p cos (vs t — as}-
3 = 1
A glance at Fig. 37 shows us that, as we increase n, the ratios
at least of the smaller frequencies approach those of the integers,
1, 2, 3, .... By passage to the limit we may demonstrate that this
is exactly true for all the frequencies.
If Q be the line density of matter of the continuous string, that
is, the mass per unit length, we have
Accordingly since
we have in the limit
Ql*
Introducing this into the value of vtj 92),
•\c\-\\
^
46]
GRADUAL PASSAGE TO LIMIT.
169
As n increases without limit y preserves its form, while vs approaches
the limit
102) ".-
We have therefore for the continuous string,
y =
• STtX
s sm ~ cos
_
fsTf-l /S \
(r v 7 ' i ~ "•) -
The frequencies of the different terms of the series are in the ratios
of the integers. Such partial vibrations are called harmonics or
overtones of the lowest or fundamental, for which s = 1. Since, if
we consider a single term of the series, the excursions of all the
particles are in the same ratios throughout the motion, we see that
the harmonics are normal vibrations. On account of the factor
depending upon x the sth harmonic has nodes for
I 2Z (s-l)Z
x = —>—,••• 2 - '->
s s s
or at any instant the string has the form of a sine curve and is
divided by nodes into s segments vibrating oppositely, generally
known as ventral segments.
In order to show how rapidly the string of beads approximates
to the motion of a continuous string, the following table from
Rayleigh's Theory of Sound is inserted. It is to be noticed that it
does not give exactly the ratios of the frequencies on account of the
variable factor s under the sine in vt> but it approximately does so,
and for the fundamental, s = 1, it gives exactly the ratio of frequency
for n beads to that of the continuous string.
n
1
2
3
4
9
19
39
2(n+l) . it
-9003
.9549
• 9745
•9836
•9959
-9990
•9997
* ' n2(^+1)
By means of an extension of the above method, Pupin has treated
the problem of the vibrations of a heavy string loaded with beads,
1) Writing the factor of — I/ — in the form
S7C \
: h
S7t
Q
S7C
since
,.
lim
= 1,
we obtain the result.
170 V. OSCILLATIONS AND CYCLIC MOTIONS.
both for free and forced vibrations, and by an electrical application
has solved a very important telephonic problem.1)
On account of the importance and typical nature of the problem
of the continuous string, we shall also solve it by means of Hamilton's
Principle. Replacing the length of a segment a by the differential dx,
writing gdx for the mass m, and for yY-of (partial derivative because
y depends upon both t and x), and for the sum, the definite integral,
we have the kinetic energy
104) r =
Similarly in the potential energy the limit of the term
1S
so that the potential energy becomes
105) W
As the number of degrees of freedom is now infinite we are
not able to use Lagrange's equations, but we can use Hamilton's
Principle, which includes them.
106)
to 0
Integrating the first term partially with respect to t and the second
with respect to x,
i t,
107)
The variation dy is as usual to be put equal to zero at the time
limits, and, as the ends of the string are fixed, dy equals zero at
1) Pupin, Wave Propagation over non - uniform Electrical Conductors.
Trans. American Mathematical Society, I, p. 259, 1900.
46] PARTIAL DIFFERENTIAL EQUATION. . 171
the limits for x also, consequently we must have the factor of the
arbitrary 6y vanish, that is,
108)
tting
motion of the continuous string,
o
Putting •— = a2 we have the partial differential equation for the
which may also be obtained from the ordinary differential equa-
tions 84) by passage to the limit in an obvious manner.
The passage from n ordinary differential equations to a single
partial differential equation when n is infinite is worth noting as a
type of a phenomenon of frequent occurrence. At the same time the
notion of normal vibrations gives rise to that of normal functions.
To find a normal vibration let us find a particular solution of 109),
110) y = X(x) • x-dw = afd^-
Dividing by Xcp we have
1 d*y _ a* d*X
~^ dt* ~X~di*"
Since one side depends only on x and the other only on t, which
are independent variables, this can hold only if either member is
constant, say — v2a2, where v is arbitrary. Thus we have the two
equations
„•«* =
112)
The first of these shows, like 77), that (p is a normal coordinate.
Its integral is
113) (p = C cos (vat — a),
the integral of the second is
114) X = A cos vx + Bsinvx.
The normal vibration is accordingly represented by
115) y = (Acosvx + B sin vx) COB (vat — a),
the arbitrary constant C being merged in A and B.
172 V. OSCILLATIONS AND CYCLIC MOTIONS.
Since for all values of t, y = 0 for x = 0, we must have J. = 0,
and since y = 0 for x = l, we must also have JE?sini'? = 0, that is
116) vl = sit,
where s is any integer, accordingly we obtain for the sth normal
vibration,
117) v. - ?,
and the vibration is given by
., ON -r, . Stt# (STtat \
118) y = ft sm -j- cos ( — ^ asj .
The general solution is therefore represented as an infinite series of
normal vibrations,
^ON -n • fSTtat
103) y =2^ J5, sm —— cos
s=l
the arbitrary constants, Bs, as, being determined by the initial dis-
placements and velocities. In order to determine them let us make
use of the other fundamental property of normal coordinates, namely,
that the energy functions do not contain product terms. Let us write
119)
then
120)
0 0
I I
= I
n A I . snx . ritx -,
f(x) sm -y- dx = ^ As I sin — sin ~- dx,
o o
and by the property just found the integral on the right vanishes in
every term except that in which r = s. But
i
I
o
Therefore we have the value of the coefficient
i
124) Ar = jff(%) sin ^dx.
o
We are thus led to a particular case of the remarkable trigono-
metric series associated with the name of Fourier. Such series were
first considered by Daniel Bernoulli in connection with this very
problem of a vibrating string. This determination of the coefficients
was given by Euler in 1777. The importance of the series in
analysis was first brought out by Fourier who insisted that such a
series was capable of representing an arbitrary function, as had been
maintained by Bernoulli, but doubted by Euler and Lagrange.
47. Forced Vibrations of General System. Let us now
briefly consider the question of forced vibrations of the general
system of § 45.
Suppose that there is impressed upon each coordinate a harmo-
nically varying force,
Fr = Ercospt,
the period and phase being the same for all, the amplitude Er being
taken at pleasure. The equations are most easily dealt with if, instead
of proceeding as we did in treating equations 41) and 42) we make
use of the principle that, in an equation involving complex quantities,
the real and the imaginary parts must be equated separately. Let
174 V. OSCILLATIONS AND CYCLIC MOTIONS.
us therefore put instead of the above value of Fr the value
whose real part agrees with the above, and having found a particular
solution of the differential equation, let us retain its real part only.
Thus we have instead of equations 63) n equations of which the rth is
d22i d*qz d*qn
125) arl i + #r2 2 ~^~ ' l~ arn
dq, dq, i dqn
+Cr2qz H ----- h crnqn
Guided by the result of § 44, assuming
these become
(
126) ;
If we call the determinant of equation 65) -D(A) and the minor
of the element of the yth column and sth row Dr»(X), we have as
the solution of 126)
127)
Since D(Ji) = 0 is the determinantal equation 65) for the free vibra-
tion, whose roots are At, A2, . . . fan, we have
128)
where 0 is the proper constant.
Accordingly the denominator D(ip) is
129) D(if) = C (V - V = (- ft + < (P - ".))•
«=1 «=1
The minors Drs(ip) are polynomials in ^ and the numerators are
therefore complex quantities, which however reduce to real ones if
the jc's are zero. We may write
130)
where Sr and &r are real and &r vanishes with the sc's, and is small
if they are small. We thus have
47, 48] FORCED VIBRATIONS. 175
131) Ar = -
X ' s = n
where
P, = >V + G>-vOa, tan «, = -—*.
n»
Retaining now only the real parts, we have for our solution,
— ;
Thus if the damping coefficients ^ are small, all the oscillations
are in nearly the same phase. If the frequency of the impressed
force coincides with that of any one of the free oscillations, p — i>s = 0,
and one factor of the denominator reduces to ^is) so that if the
damping of that oscillation is small, the amplitude is very large, or
infinite if there is no damping. This is the case of resonance.
(Resonance may also be defined in a slightly different manner as
occurring when ip is one of the roots of the equation D(T) = 0 in
which all the «'s have been put equal to zero. This corresponds
with our example in § 44. In practical cases the difference is very
small.)
48. Cyclic Motions. Igiioratioii of Coordinates. In certain
large classes of motions some of the coordinates do not appear in
the expression for the kinetic energy, although their velocities may.
For instance in the case of rectangular coordinates,
the coordinates themselves x, y, z do not appear. In spherical co-
ordinates, § 41, 133),
(p does not appear while both r and # do. Further examples are
furnished in the case of systems in which throughout the motion
the place of one particle is immediately taken by another equal
particle moving with the same velocity, as for instance in the case
of the system of balls in a ball-bearing (bicycle) or better in the
case of a continuous chain passing over pulleys, or through a tube
of any form, or by the particles of water circulating through a tube.
176 V. OSCILLATIONS AND CYCLIC MOTIONS.
In order that this condition may be permanent it is evidently
necessary that the path traversed by the successive particles shall be
reentrant, or that they shall circulate. Under the conditions supposed
it is evident that the absolute position of any particle does not affect
the kinetic energy, for throughout the motion at any point on the
path of the particles there is always a particle moving with the same
definite velocity. On account of the character of these examples the
term cyclic coordinates has been applied by Helmholtz to coordinates
which do not appear in the kinetic energy. We shall when necessary
distinguish cyclic coordinates by a bar, thus
133) |£ = 0
tig.
is the condition that q^ is cyclic. This of course involves that every
that is the coefficients of inertia do not depend upon the cyclic co-
ordinates. Thus a cyclic coordinate is characterized by the fact that
the corresponding reaction is wholly momental. Examples of cyclic
coordinates are found in x, y, #, qp, above, and cp in the case of plane
polar coordinates.
Inserting equation 133) in Lagrange's equations we have
ia^ d i
dt(
or the fundamental property of a cyclic coordinate is that the force
corresponding goes entirely to increasing the corresponding cyclic
momentum. If the cyclic force Pr vanishes, we have
and integrating,
iw\ VT
W,r = Pr = Cr.
In this case we may with advantage employ a transformation intro-
duced by Routh1) and afterwards by Helmholtz2), which is analogous
to that invented by Hamilton and described in § 39. By means of
equations 53) and 71) § 39, we have expressed the velocities as
linear functions of the momenta with coefficients Brs, which were
functions of the coordinates, and have thus introduced the momenta
into the kinetic energy in place of the velocities. We have thus
been led to use instead of the Lagrangian function L = T — W,
1) Routh, Stability of Motion, 1877.
2) Helmholtz, Studien zur Statik monocydischer Systeme, 1884. Ges. Abh.
Ill, p. 119.
48] ELIMINATION OF VELOCITIES. 177
whose variation appears in Hamilton's Principle, the Hamiltonian
function H=TJrW. The transformation of Routh and Helmholtz,
instead of eliminating all the m velocities q', eliminates a certain
number, which we will choose so as to replace those having the
suffixes 1, 2, . . . r, by the corresponding momenta, but to retain the
velocities with suffixes r + 1, . . . w, in the equations. This trans-
formation, while it may be made in the general case, is of particular
advantage where the eliminated velocities are cyclic and the corre-
sponding momenta constant, as in the case just described.
The equations 53) § 39 for the elimination become by trans-
position
138)
Qriqi + GrSffi +• • '+ Qrrq'r = Pr ~ (fc.r + ltfr'+l +' ' ' + Crmffi).
It will be convenient to write the right hand members above,
Pi-Si, ...pr-Sr.
Let the solutions of equations 138) be
&' = -Rll (Pi - $) + R** (P-2 ~ SJ + • • + Rlr (Pr ~ &),
139)
qr' = Rrl (Pi ~ St) + Bri (pi - S2) + • • + Err (pr ~ &),
where the J^'s are the quotients of the corresponding minors of the
determinant
Qn> 612? • • • Q
Qrl) Qr2, - • • Qrr
by the determinant itself, and, like the §'s, are functions of the
coordinates only.
Introducing the values 139) for the q"s into the kinetic energy,
the latter becomes a function of the velocities qr+i, - - • q™ and of
the momenta pl9 .. .pr. It is a homogeneous quadratic function of
all these variables, but not of the p's or g"s considered separately
on account of product terms, such as psqt' which are linear in terms
of either the pjs or q"s. The function I thus transformed has lost
its utility for Lagrange's equations, but may be replaced by a new
function, as follows.
Let us call the function T expressed in terms of the new
variables T'. We have thus identically
140) T(ql9 q2>... qm, &', &', ...$!»).
WEBSTER, Dynamics. 12
178 V. OSCILLATIONS AND CYCLIC MOTIONS.
It is to be noticed that since the coordinates q appear in the co-
efficients E of equations 139) they are introduced into T' in a way
in which they do not appear in T, so that we do not have
dT dT'
hut since q enters in T' both explicitly and implicitly through
equations 139), we have for s = r -f 1, r + 2, . . . w,
141) - =
3
is called by Routh the modified Lagrangian function, and its negative
_by Helmholtz the kinetic potential. It is to be understood that 0 is
to be expressed in terms of the velocities , 5/4-1, . . . q'n by means of
equations 139) in which plf . . . pr have been replaced by C1; . . . cr.
The important thing to notice about
like the Q's from which they are derived. The terms of the latter
sort in — -rr— cause precisely the same effect as if they were added
to the potential energy. The effect of cyclic motions in a system is
accordingly partly represented by an apparent change of potential
energy, so that a system devoid of potential energy would seem to
possess it, if we were in ignorance of the existence of the cyclic
motions in it. The effect of the linear terms in 0 is quite different
and will be discussed in § 50.
A system is said to contain concealed masses, when the coordinates
which become known to us by observation do not suffice to define
the positions of all the masses of the system. The motions of such
bodies are called concealed motions. It is often possible to solve the
problem of the motions of the visible bodies of a system, even when
there are concealed motions going on. For it may be possible to
form the kinetic potential of the system for the visible motions, not
containing the concealed coordinates, and in this case we may use
Lagrange's equations, as in the case just treated, for all visible
coordinates, while the coordinates of the concealed masses may be
ignored. Such problems are incomplete, inasmuch as they tell us
nothing of the concealed motions, but very often we are concerned
only with the visible motions. Such concealed motions enable us to
explain the forces acting between visible systems by means of
concealed motions of systems connected with them.
The process of eliminating the cyclic coordinates of the concealed
motions as above described is termed by Thomson and Tait ignoration
of coordinates.1)
Examples of the process may be obtained in any desired number
from the theory of the motion of rigid bodies rotating freely about
1) Thomson and Tait, Natural Philosophy, Part I, § 319, example G.
12*
Y- OSCILLATIONS AND CYCLIC MOTIONS.
axes pivoted in bearings fastened to bodies themselves in motion.
Such motions will be treated in § 94.
A very simple case of the above process is encountered in
treating the motion of a particle m sliding on a horizontal rod,
revolving about a vertical axis, at a distance r from the axis. Let
the angle made by the rod with a fixed horizontal line be cp, then
the velocity perpendicular to the rod is rep'. The velocity along the
rod being rf, the kinetic energy of the body m is
146) T=~
Since (p does not appear in T, (p is a cyclic coordinate. If there is
no force tending to change the angle y> we have
dT
147) P(p = — = mr*
mentioned, ~mr'2 being the
quadratic function of the remaining velocity r' and — — ^— g being
the quadratic function of the constant c, which contains as a coeffi-
cient a function of the coordinate r. We may now, ignoring the
coordinate
or
dr
1R1\ r $ c
mW = ^ = ^r*'
We accordingly see that the system acts as if, there being no rotation,
it possessed an amount of potential energy — C&, producing the force
s>2
^3 directed from the center. This example accordingly illustrates
the effect of ignored cyclic motions in producing an apparent potential
energy, but it does not illustrate the effect of linear terms in
* + &»&" *
If qB is the cyclic coordinate, all the Q's are independent of g3, and
if the corresponding force P3 vanishes, we have the constant momentum,
153) pB = Cis &' + fts &' + £33 &' - ^3 ;
From this we determine the cyclic velocity,
.!«) ft,_^-ft.«/-ft.&',
inserting which in the kinetic energy gives, on combining terms,
155) T'=±
+ -«-
It is noticeable that the linear terms in #/, q2' have cancelled each
other. It will be proved below that this always happens. But when
we form the kinetic potential, which is to be used instead, they
reappear. We have
156) = T'-
)
V33
, csQ13 , csQ, , 1 c32
I O bfl~i/0 b/2 ^/O
V33 VSS ? V33
Thus the effect of the cyclic motion, which may itself be concealed
from us, is made evident to our observation by the presence of the
fourth and fifth terms, which are linear in #/, q2r. The apparent
coefficients of inertia, that is the cofficients of qt' 2, q2' 2, #/ g2'; are
182 V. OSCILLATIONS AND CYCLIC MOTIONS.
changed from their real values (unless $13 — $23 = 0), while
c 2
there appears the term - ~— independent of the velocities, depend-
r Vss
ing on the coordinates qlf q2. This is, since it gives rise to a
conservative positional reaction, undistinguishable in its effect from
potential energy. In reality, the reaction to which it gives rise is
motional, instead of positional, as it appears to be. If we could
explain all potential energy in this manner, namely as due to concealed
cyclic motions, we should have solved the chief mystery of dynamics.
In his remarkable work on dynamics, Hertz treats all energy from
this kinetic point of view. In order to have a successful model for
this representation of potential energy, which needs in order to be
perfect no linear terms, we must have Q13 = Q23 = 0.
We can now see why the simple example of § 48 showed no
linear terms, since by putting all the Q's with one suffix 2 equal to
zero we pass to the case of a system with two degrees of freedom.
If at the same time the coordinates are orthogonal, §13 = 0, so that
the single linear term disappears. This was the case above.
Let us now pass to the general case. We have for the momenta
the equations 53) § 37 and, for the first r, 137) which are written out,
Pi =
' 2 m
157)
Pr =Qr
Pm — Qml (
Let us now form the kinetic energy from the definition, § 36, 38),
158)
Multiplying the above equations, the sth line by qa', and adding, we
obtain from the first r lines on the right,
The terms coming from the last m — r lines, and the first r columns,
as marked off by the dotted lines, are found to be, on collecting
according to columns,
49] EFFECT OF ELIMINATION. 183
' s = r
2*'*'
«=i
on referring to tlie definition of the definitions of the 5s's, 138),
159) S, = Qs,r
Finally the terms from the lower right hand square, of m — r rows
and columns gives us a quadratic function of the last m — r velocities,
namely that part of 2T which originally depended on these velocities
and no others. This part we will call 2Ta. We have therefore
160) 2T = 2Ttt + g.' (8. + e,\
5 = 1
Now if we form the quadratic functions, with the coefficients R from
the determinant of equations 139),
s =• r t = r
= -£•/, /, KitSt
we may write equations 139) as
lea) /=!£-!! '(.- 1,2, ..,-),
s s
so that we may write
s = r
163) 2T = 2Ta +2 (c, + S,) g£ - g) •
s = l s s
But since (7, /S are homogeneous functions of cs, Ss respectively,
S=l * 8 = 1
so that the above becomes,
164) 2T=2Ta + 2C-2S
But we also have
so that the sums in 164) destroy each other, and there remains
165) T' = Ta-S+C.
184 V. OSCILLATIONS AND CYCLIC MOTIONS.
But S is a homogeneous quadratic function of the $/s, which are
themselves homogeneous linear functions of the g/'s, so that S, like Ta,
is a homogeneous quadratic function of the non- eliminated velocities.
Thus we have proved that the linear terms disappear from the kinetic
energy. At the same time we have obtained the general value of
the part independent of the velocities. Forming the function 0 for
the kinetic potential,
166) ^ = T'
so that the part C which imitates the potential energy is a homo-
geneous quadratic function of the momenta cs of the concealed cyclic
motions. The terms under the sign of summation are linear in the
remaining velocities.
5O. Effect of Linear Terms in Kinetic Potential. Gyro-
scopic Forces. We will now examine the effect of terms linear in
the velocities in the kinetic potential, arising from any cause what-
ever. We have seen that such terms arise from variable constraints,
and from ignored cyclic motions. We shall find a third case when
we treat of relative motion, § 103.
Suppose now that the kinetic potential contains the linear part
167) <&t = L! $1 + L2 g2' H h Lm%mj
where the coefficients L are functions of the coordinates, and may
also involve the time explicitly. Let the part of the force Ps that
must be applied on account of the part d^ be denoted by P/1), so that
168) *£*i)_aj«l = p.M.
dt\dq'J dqs
Now
and differentiating,
We have also
49, 50J EFFECT OF LINEAR TERMS. 135
Using these values in 168), we obtain for the force,
= ^sl &' -f ^2 &' + • - • + #.mgm + -fci
where the 6r's are functions of the coordinates defined hy
For the force applied to change a coordinate qt we have a similar
form, with coefficients such that
IT*) . ' *,.-£-£ — *.,.
We have then the result that the terms linear in the velocities in
the kinetic potential give rise to reactions linear in the velocities,
with the property that the coefficient of q} in the reaction Ps is
equal and of opposite sign to the coefficient of qj in the reaction Pt.
Such reactions are called gyroscopic forces by Thomson and Tait1),
since we have examples of them where gyrostats, or symmetrical
bodies spinning about axes attached to parts of systems, act as
concealed cyclic motions. If we find the activity of the gyroscopic
forces,
173) d- =
we find that in the part PSW qs' we have the term G-stqs' m. the equation of activity. These forces are
consequently conservative motional forces. They are however perfectly
distinguishable by their effects from the conservative motional forces
arising from the term C which imitates potential energy, and they
in no wise imitate potential energy, as we shall see by an example.
A system containing gyrostatic members behaves in such a peculiar
manner that their presence is easily inferred. The theory of gyro-
stats will be treated in Chapter VII. In the mean time the following
simple example will illustrate the theory, and at the same time serve
to prepare for the general theory of the gyrostat, of which it con-
stitutes a special case.
1) Thomson and Tait, Nat. Phil. § 345^1.
186
V. OSCILLATIONS AND CYCLIC MOTIONS.
Let four equal masses, —> be fastened to the ends of two mutu-
ally perpendicular arms of negligible mass (Fig. 38), which are fastened
rigidly where they cross, at
their middle points, to an axis
perpendicular to them both,
about which they turn. Let the
point of crossing of the three
arms be fixed while the system
can spin about the axis OP,
which can move in any manner.
We will suppose that during
the motion the axis OP makes
with the ^-axis a small angle
whose square can be neglected
in comparison with unity. Let
the position of the axis be
determined by the coordinates
!, 77, of the point in which it
intersects a plane perpendicular
to the ^-axis at unit distance
Fig. as. from the origin. The squares
and products of |, y, are con-
sequently to be neglected. Let us further specify the position of the
system by the angle cp that the projection of the arm OA on the
XY- plane makes with the X-axis. Thus the three coordinates |, rj, cp
determine the position of the whole system.
If the coordinates of the point A are x, y, 8, since it lies in a
plane whose normal passes through the point £, ??, 1, we have
174) g + lx 4- yy = 0.
But since OA always makes a small angle with the XT- plane, the
projection of OA on this plane differs from it in length only by a
quantity of the second order, which we neglect. We therefore have
Differentiating 174),
dy = xdcp.
so that we have
dx2 +
= xdl -f ydri -f (rjx — ly) dcp,
+ dz* = (Z2 + rfx* + |y -
50] GYEOSCOPIC FORCES. 187
and the part contributed by the particle A to the kinetic energy is
The opposite particle C, for which x2, y2, xy have the same values,
contributes the same amount. The other pair of particles, for which
the values of x2, y2 are respectively those of y2, x2, for the first pair,
and the values of xy the negatives of the values for the first pair,
consequently contributes an amount of energy which, added to that
already found, makes the terms in xy disappear, and replaces each
term in x2, y2, by the same term with I2 written in the place of x2
or y2. Neglecting then |2, if, we have finally
176) y='
We accordingly see that cp is a cyclic coordinate for the system, so
that if the system is spinning without any force tending to change qp,
we are dealing with a case of the example in § 49. We have, pro-
ceeding as there,
177)
and eliminating qpf,
from which we form
In order to form the diiferential equations for the motion of |, 77,
we have by differentiation
179)
and neglecting the squares and products of the small quantities |,
and £', ?/, which are small at the same time,
w
a*
ai
188 V. OSCILLATIONS AND CYCLIC MOTIONS.
Proceeding in the same manner for 77, we have with the same
degree of approximation
d$ ml* , c£
W = "~^n "V
181)
If W is the potential energy (there being no apparent potential
energy due to the cyclic motion, since the part C is here constant),
the equations of motion are accordingly,
182)
Thus the gyroscopic terms in c have the property proved in 172).
If there is no potential energy, the gyroscopic forces cause the
motion to be of such a nature that
rr + tf'i/ = o, i/i"2+v12 - ^v^+^'
that is the acceleration is perpendicular to the velocity, and pro-
portional to it. Under these circumstances the motion is uniform
circular motion. In fact the equations are satisfied by
I = Acospt. vc
183) W = 0, p = ™-
t] = A srnpt, ml*
Thus the circle, whatever its size, is described in the same time
~> which is inversely proportional to the momentum of the cyclic
motion. We may describe the effect of the gyroscopic forces in
general for a system with two degrees of freedom by saying that
they tend to cause a point to veer out from its path always toward
the same side. This effect is characteristic, and cannot be imitated
by any arrangement of potential energy whatever. By the aid of
this principle all the motions of tops and gyrostats may be explained.
51. Cyclic Systems. A system in which the kinetic energy
is represented with sufficient approximation by a homogeneous
quadratic function of its cyclic velocities is called a Cyclic System.
Of course the rigid expression of the kinetic energy contains the
velocities of every coordinate of the system, cyclic or not, for no
mass can be moved without adding a certain amount of kinetic
energy. Still if certain of the coordinates change so slowly that
their velocities may be neglected in comparison with the velocities
of the cyclic coordinates, the approximate condition will be fulfilled.
These coordinates define the position of the cyclic systems, and may
50, 51] CYCLIC SYSTEMS. 139
be called the positional coordinates or parameters of the system. In
the example of § 48 if we suppose the radial motion to be so slow
that we may neglect rn in comparison with r2cp'2 we have
184) T=ymrV2,
and the system is cyclic, r being the positional, cp the cyclic co-
ordinate. In the case of a liquid circulating through an endless
rubber tube, the positional coordinates would specify the shape and
position of the tube. The positional coordinates will be distinguished
from the cyclic coordinates by not being marked with a bar. The
analytical conditions for a cyclic system will accordingly be, for all
coordinates, either
1Q.s 3T n 3T
18o) ^ = 0 or ^7=^ = 0,
or if we use the Hamiltonian equations 78) § 39 with the value of T
obtained by replacing the velocities by the momenta, which we shall
denote by Tp, since the non- cyclic momenta vanish
186) ^ = 0, and |5? = 0,
for the cyclic coordinates, as before. We accordingly have for the
external impressed forces tending to increase the positional coordinates,
by § 37, 60), § 39, 80) respectively, the first term vanishing,
TT) _d(lP+W) i)
dT
w- P-
s~
and for the cyclic coordinates
A motion in which there are no forces tending to change the
cyclic coordinates is called an adiabatic motion, since in it no energy
enters or leaves the system through the cyclic coordinates. (It may
do so through the positional coordinates.) Accordingly in such a
motion the cyclic momenta remain constant. The case worked out
above was such a motion.
In adiabatic motions the cyclic velocities do not generally remain
constant. In the above example, for instance, the cyclic velocity (p'
was given by
A motion in which the cyclic velocities remain constant is called
isocydie.
o rn O rji
1) That - - = -o-^ may be seen by putting r = m in 144) , when the
parenthesis becomes T' — 2T=—TP.
190 V. OSCILLATIONS AND CYCLIC MOTIONS.
The motion of a particle relatively at rest upon the surface of
the earth is isocyelic, taking account of the earth's rotation.
In such a motion the cyclic momenta do not generally remain
constant, but forces have to be applied.
In the example of the bead on the revolving rod if r varied
forces would have to be applied to the rod to keep the rotation y the cyclic forces
is double the work done by the system against the positional forces.
In such motions the energy of the system accordingly increases by
one-half the work done by the cyclic forces, the other half being
given out against the positional forces. For if we use the energy
in the form
we have in any change
196) dT = \^l (qj dps + ps dfr'),
and in an isocyclic change, every dqs' vanishing,
197) ST- ±2 *•'**•
But since
198) ^ = P., dj>. = Psdt, and since qi = -~±> ql dt = dqs,
and the above expression for the gain of energy becomes
199) dT =
But the work done by the cyclic forces is
200) 8 A =?sPsdqs = 2dT.
Therefore the last part of the theorem is proved. Again, in any
motion,
sol) w
and in an isocyclic motion,
202) . 8T
But since the work of the positional forces is
203) 8 A = p. 8q, _ - dq. = - ST,
the first part of the proposition is also proved.
II. In an adiabatic motion, the cyclic velocities will in general
be changed.
Then they change in such a way that the positional forces
caused by the change of cyclic velocities oppose the motion, that is,
do a positive amount of work. For since for any positional force
53, 54] WORK DONE BY FORCES. 193
P ar
p*~ 3£;
the change due to the motion is
Of this the part due to the change in the cyclic velocities is
205) • tP.
and the work done by these forces is
206) S-, A ~£. 9? P.Sq, - -
Now we have for any motion
207)
and in an adiabatic motion this is .zero., so that
208)
Substituting this in the double sum 206) r we get
209) SA
But this expression represents [§..36, 35)] twice the energy of a
possible motion in which the velocities would be dqt', and must
therefore be positive for all values of dqj, dgr'.
Accordingly d- A > 0.
The interpretation of this theorem for electrodynamics is known
as Lenz's Law1), namely, ato, electrical current being represented by
a cyclic velocity, and the shape and relative position of the . circuits
by positional coordinates, if in any system of conductors carrying
currents, the relative positions of the conductors are changed, the
induced currents due to the motion of the conductors are so directed
as by their magnetic action to oppose the motion.
54. Examples of Cyclic Systems. Let us consider the example
of equation 184) as illustrating the previous theorems.
We have for the momenta
dT dT
1) These Theorems are all given by Hertz, Prinzipien der MechaniJc,
568 — 583.
WEBSTER, Dynamics. 13
194 V. OSCILLATIONS AND CYCLIC MOTIONS.
and introducing these instead of the velocities
210) ^
We have for the positional force
This being negative denotes that a force Pr toward the axis
must be impressed on the mass m in order to maintain the cyclic
state. This may be accomplished by means of a geometrical constraint,
or by means of a spring. The force or reaction — Pr which the
mass m exerts in the direction from the axis in virtue of the rotation
is the so-called centrifugal force. We see that if the motion is iso-
cyclic, the positional force increases with r, while if it is adiabatic,
as in the case worked out above, it decreases when r increases. The
verification of the theorems of § 52 is obvious. The cyclic force
vanishes when the rotation is uniform, and the radius constant. If,
the motion being isocyclic, that is, one of uniform angular velocity,
the body moves farther from the axis, Pv> the cyclic force is positive,
that is, unless a positive force P9 is applied, the angular velocity
will diminish. In moving out from r: to r2 work will be done
against the positional force Pr of amount
r2 rz
212) - A = -Jprdr = my' *Jrdr = ^ (r22 - r*)9
^ rj,
while the energy increases by the same amount.
Thus the first theorem of § 53 is verified. If the motion is
adiabatic,
o f
If the body moves from the axis, cp' will accordingly decrease,
so that
213)
The change in Pr due to a displacement dr is, by 211),
214) dPr = - m(yndr + 2/V (V),
of which the part containing dcp',
215)
does the work
54] EXAMPLES OF CYCLIC MOTIONS. 195
216) dv'A = d9'Prdr =
or by 213),
217) d(p'A = mr*d
we see that the
rod carrying the particles will remain at rest relatively to the hori-
zontal rod in either a vertical or horizontal position. It is easy to
see that the vertical position is one of unstable equilibrium, for,
writing the equation 209)
220)
we see that if # be slightly different from zero, # will tend to become
still greater in absolute value. Writing however # = -y — &' the
equation becomes
221)
1) The system is cyclic if we neglect
13*
196 V. OSCILLATIONS AND CYCLIC MOTIONS.
If #' is slightly different from zero, it will accordingly tend to
approach the value zero, so that the horizontal position is stable.
A body moving according to the differential equation 221) is
called by Thomson and Tait1) a quadrantal pendulum, since # changes
"according to the same law with reference to a quadrant on each
side of its position of equilibrium as the common pendulum with
reference to a half -circle on each side", or in other words, in the
ordinary pendulum the acceleration is proportional to the sine of
the angle of deviation from equilibrium, and in the quadrantal to
the sine of twice the angle. The small oscillation performed by the
bar will be harmonic with the frequency ~- Here we have an ex-
cellent example of an apparent potential energy which is really kinetic.
1) Thomson and Tait, Nat. Phil., § 322.
PART II
DYNAMICS OF RIGID BODIES
55]
TRANSLATIONS AND ROTATIONS.
199
CHAPTER VI.
SYSTEMS OF VECTORS. DISTRIBUTION OF MASS.
INSTANTANEOUS MOTION.
55. Translations and Rotations. A rigid body or system
of material particles is one in which the distance of each point of
the system from every other is invariable. Its position is known
when the positions of any three of its points are known, for every
point is determined by its distances from three given points. These
three points have each three coordinates, but, since there are three
conditions between them, defining their mutual distances, there are
only six independent coordinates. Thus, a rigid body has six coordinates.
A rigid body may evidently be displaced in such a manner that
the displacement of every point is represented by equal vectors, that
is equal in length and parallel. Such a dis-
placement is called a translation, and, being
represented by a free vector, has three coordinates.
A rigid body may also evidently be displaced,
so that two given points in it, A and IB, remain
fixed. Since any point P must move on a sphere
of radius BP about B, and also on a sphere of
radius AP about A, the locus of its positions is
the intersection of the two spheres, that is a circle
whose plane is perpendicular to the line AB, and
whose radius CP is the perpendicular distance
from P to the line AS. If this is zero, the
point does not move, therefore all points on the
line AB remain fixed. The displacement is called
a rotation and the line AB, the axis of rotation. The rotation is
specified if we know the situation of the line AB and the magnitude
of the angle POP', or the angle of rotation.
A line may be specified by giving the two pairs of coordinates
of the points in which it intersects two of the coordinate planes.
A line has thus four coordinates, and a rotation, five, — the four
of the axis together with the magnitude of the angle.
Fig. 39.
200 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
Any displacement of a rigid body may be brought about in an
indefinite number of ways. Let three points ABC (Fig. 40) be displaced
to A'B'C'. We may first
give the body a translation
defined by the vector A A'.
This will bring B to B± and
C to Cv Then through A'
pass an axis perpendicular
to the plane B^B', and
rotate the body about this
axis through the angle B^A'S1.
This brings B± to B' and C±
to a new position (72. Finally
rotate the body about A'B'
until C2 arrives at Cf. We have thus brought about the given dis-
placement by means of a succession of translations and rotations.
Evidently the order of these may be varied. Accordingly,
Any displacement of a rigid body may be reduced to a succession
of translations and rotations.
We have seen that a translation may be represented by a free
vector, a rotation, by a vector that must give the axis and the angle.
If we agree to draw the vector in the axis, and make its length
numerically equal to the angle of rotation, it will completely specify
the rotation, if we adopt a convention about the direction of rotation.
This shall be that, if the rotation is in the direction of the hands
of a watch, the vector shall point from face to back of the watch.
Vector and rotation correspond then to the translation and rotation
in the motion of a cork-screw, or any right-handed screw. As the
vector may be placed anywhere along the axis, but not out of it,
it has five coordinates, and may be characterized as a sliding vector.
Translations are compounded by the law of addition of vectors.
The resultant of two rotations about the same axis is evidently the
algebraic sum of the individual rotations. The resultant of a trans-
lation and rotation is evidently independent of the order in which
they take place.
The resultant of a rotation and a translation perpendicular to
its axis is equivalent to a rotation about a parallel axis, for it is
evident that all points move in planes perpendicular to the axis, and
that the motions of all such planes are alike, or the motion is
uniplanar.
Now the motions of any two points in a plane determine the
motion of the plane parallel to itself.
55, 56]
ROTATIONS ABOUT PARALLEL AXES.
201
From 0 (Fig. 41) lay off the translation vector 00' of length, t and
find a point C on the perpendicular bisecting 00' which makes the
angle OCO' equal to ca, the angle of rotation, and in the right sense.
Then if OC be rotated about 0 through the angle to to Cf and
then C" be moved by the translation it will
return to C. Therefore the point C remains
fixed, and is the center of rotation, and
thus the rotation co about C is equivalent
to the equal rotation about 0 together with
the translation,
1) r = 200 sin ~,
and if p is the perpendicular from C to 00',
a 2 Fig. 41.
56. Rotations about two Parallel Axes. As before the
motion is uniplanar and is specified by two points. Let A and B (Fig. 42)
be the intersections of the axes with the plane of the paper perpen-
dicular to them. Turn about
A through the angle co1,
bringing B to B'. Then turn
about B' through the angle a>2,
bringing A to A'. Bisect o1
by AC. B could be brought
to B' by rotation about any
point of AC, since all such
points are equidistant from
BB!. Bisect w2 by B'D.
A could be brought to A'
by rotation about any point
in B'D. Therefore the motion
of A and B could be produced
by a rotation about 0, the intersection of AC and B'D. Triangle
AOA' is isosceles.
Angle AOD = angle OAB' + angle AB'O = ^ + ^,
Angle AOA' =-- 2 - angle AOD = o^ + o>2,
that is, two rotations about parallel axes compound into a rotation
equal to their algebraic sum about a parallel axis. To find the
position of this axis we have
OS' OA AB
sm— '-
—*- sin
2 ~ 2 2
If the order of rotation is changed we obtain a different result.
202 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
If the rotations c^ and o2 are of opposite signs and of equal
magnitudes, the intersection of the two bisectors is at infinity and
the axis of rotation is thus at infinity. A motion about an infinitely
distant axis is a translation. The direct proof is as follows.
Let A be the center of rotation a , bringing B to B'. Then
rotate about B' through an equal angle in the opposite direction,
bringing A to A. Triangles
ABB' and AAB' have AB'
common, and AB = A'B' and
the included angles equal,
therefore A A' and BB' are
equal and parallel and two
points consequently all
points — have moved parallel
to each other the same distance. The motion is therefore a trans-
lation of magnitude,
4)
. 43.
t = 2AB sin ~
Accordingly every translation may be decomposed into rotations, and we
may reduce all displacements to rotations.
57. Rotations about Intersecting Axes. Infinitesimal
Rotations. Let OA and OB be two intersecting axes about which
we revolve the body through the
angles c^ and o>2 respectively.
Describe a sphere with the center 0.
Let the rotation o1 about A bring
B to Bf, and o>2 about B' bring
A to A. Pass planes through
the vertices bisecting the angles oc^
and «2, then, as in § 56, the
displacement just given is equi-
valent to a rotation about the line
of intersection CO of these planes.
The order of the rotations affects
the result.
Since AC bisects the angle
BAB' and the spherical triangle
Fig. 44.
BAB' is isosceles,
angle ABC = angle AB' C = y-
Thus the resultant rotation, o = angle ACA' = angle BCB'.
Angle ACE= angle B1 CD = angleDOJ5= |-
56, 57]
RESULTANT OF INFINITESIMAL ROTATIONS.
203
In the spherical triangle ABC we have
. 0>a . 2 are laid off anywhere on their
axes, the position of the axis 0 may be found by the following
construction. At A a point
on the axis of rotation CDI lay
off A E = o2 and at I? at a
point on the axis of rotation co2
in the opposite direction BS=av
Join E and S, and where this
straight line ES cuts AB,
draw OT parallel to AE, BS
equal in length to 04 + «2. For
0 A _ AE _ co2
~OB~ B~S~~^L'
as required by 7).
The construction (Fig. 47) shows that if co1 and o>2 have the
same sign, the resultant co1 + o>2 has its axis 0 between A and B.
If co1 and o2 are of opposite signs the same construction may
be used (Fig. 48), but 0
is on AB produced and
on the side of the greater
rotation. If o^ = — o>2
,yf\^ evidently 0 is at infinity
and o = 0. The resultant
is then a translation per-
pendicular to the plane of
the two axes, and its
magnitude t is by 4) equal
to CDG31 times the perpen-
dicular distance between
the axes.
Fig. 47.
Fig. 48.
58. Vector - couples. A pair of equal, parallel, oppositely
directed, sliding vectors will be called a vector -couple. A rotation
vector -couple is thus equivalent to a translation perpendicular to its
plane, equal to the product of the length of either vector by the
57, 58, 59, 59 a] VECTOR -COUPLES. 205
perpendicular distance between their lines, or the arm of the couple.
This product is called the moment of the couple.
Two couples whose planes are parallel give rise to parallel
translations, and if their moments are equal, to equal translations.
Therefore a rotation -couple may be displaced without altering its
effect, if its plane is kept parallel to itself and its moment is un-
changed.
A vector -couple may then be represented by a single vector
perpendicular to its plane, whose length is equal to the moment of
the couple. Its direction will be governed by the same convention
as before, namely, the vector moment is to be drawn in such a
direction that rotation in the direction of the couple and translation
in that of the moment correspond to the motion of a right-handed
screw.
Moments will be represented by heavy vectors. The moment of
a vector -couple is a free vector, hence the composition of couples is
simpler than that of the slide -vectors themselves.
We may now state the theorem of the general infinitely small
displacement of a body as follows: The infinitely small displacement
of a body may be reduced to a translation and a rotation, or in other
words to a rotation and a rotation- couple. The choice of components
may be made in an infinite number of ways.
59. Statics of a Rigid Body. Two equal, parallel, opposi-
tely directed forces applied to a rigid body in the same line are in
equilibrium. For otherwise they can produce only distortion or
motion. Distortion is excluded according to the definition of a rigid
body. They satisfy the conditions of equilibrium, § 32, for if applied
at the center of mass they are in equilibrium, and their moments
about any point are equal and opposite. Accordingly a force applied
to a rigid body may be applied at any point in its line of direction
without change of effect. Thus forces applied to a rigid body are
not free, but are sliding vectors (five coordinates). (This is not a
property of forces, but of rigid bodies.) Forces, whose lines of
direction intersect, may be applied at the point of intersection and
compounded by the rule of vector addition.
AP £$
59 a. Parallel Forces. Force - couples. Let A&- and •?*$-
(Fig. 49) represent two parallel forces applied to a rigid body at A
and B. Introduce at A and B two equal and opposite forces AE
and BS of any magnitude in the line AB. These being in equili-
brium do not affect the system. Find the resultant of AP and AE
by the parallelogram, giving AC, also of BQ and BS giving BD.
All these forces are coplanar, therefore the lines AC and BD will
206 VI- SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
meet at E, if produced. Slide AC and BD to E, and then resolve
into components parallel to the original ones. We get EH and EJ
equal and opposite (being equal to AE and JBS), and EK equal to
AP and EL to BQ applied at E. Therefore the resultant of two
parallel forces is a parallel force equal to their algebraic sum, and
applied on a line
T EOj whose posi-
tion is to be found
as follows.
From the simi-
lar triangles,
AO
FK
BO
GL
OE
~KE
OE
LE
OE
AP'
OE
BQ
By division, since
FK=GL,
Fig. 49.
AO
BO
BQ
AP
Thus the position of the resultant of parallel forces is to be found
by the same construction as the resultant of two rotations about
parallel axes, Fig. 47.
If the two forces are oppositely directed (Fig. 50), 0 is on AB
produced, and if the forces are equal 0 lies at infinity. Accord-
ingly there is
no force that can
replace two equal,
parallel and op-
positely directed
forces not along
the same line, or
force- couple. The
distance between
the lines of direc-
tion is the arm,
and the product
of either force by
the arm is the
moment of the
couple.
Fig. 50.
We shall prove the following theorems.
59a]
THEOREMS ON FORCE -COUPLES.
207
Fig. 51.
Theorem I. A couple may be transported parallel to itself either
in its own or a parallel plane without changing its effect.
Consider the forces Pl and P2 both equal to P, applied perpendi-
cularly at the ends of AB (Fig. 51). At the ends of an equal
and parallel line A'B' apply
four equal and opposite forces
P3,P4,P5,P6, each equal to P,
which are in equilibrium. The
resultant of the equal parallel ^
forces P17 P6 is a force 2P |
applied half - way between A
and B'. The resultant of P2
and P5 is a force 2P in the
opposite direction applied half-
way between A and B. Since
ABB' A' is a parallelogram
these two points of application
coincide and the two resultants neutralize each other. We have left
the couple P3P4 equivalent in effect to the original couple.
Theorem II. A couple may be turned in its plane about its
center of symmetry without changing its effect.
Let A'B' be a line of the same length and with the same center
0 as AB, the arm of the couple, and in the plane of the couple
(Fig. 52). Apply at
A' and B' four equal
and opposite forces
in equilibrium, each
equal to P, and
perpendicular to
A'B' and in the
plane of the couple.
Consider Pl and P5
applied at (7, their
point of inter-
section, and by
symmetry their re-
sultant will be along
OC. Similarly the
resultant of P2 and
P6 is an equal force
along OD in the
opposite direction.
These two resultants neutralize each other, leaving the couple P3P4
which has the same effect as the original couple.
208 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
Theorem III. A couple may be replaced by another in the
same plane having equal moment.
Let the couple be P^P^ and
the arm be A S (Fig. 53). At C
on AS produced and at S apply
four equal and opposite forces Q
of such magnitude that
ft
B
AB
The resultant of the parallel
forces, P1? QB, is equal to Pl plus
Q3 applied at S on account of
the above equation. This is
counterbalanced by the forces P2
Fig. 53. and Q± applied at B, leaving the
couple Q^Qz of moment
f\ 7?/T ~p A 7?
\cJ • JD\J === JL * J* Ify
equivalent to the original couple.
A force -couple is determined therefore by its plane and moment,
and may be represented by a free vector perpendicular to its plane
and of length equal to the moment.
Theorem IV. Composition of Couples. Suppose the two couples
are in different planes. By turning each in its own plane bring all
the four forces into directions
perpendicular to the intersection
of their planes, and then by
varying one of the couples cause
them to have the same arm AS.
The forces Qjfi applied at A
compound by the parallelogram
into Ev P2 and Q2 applied at
S compound into E% equal and
opposite to Ev The arm of all
these couples is the same, there-
rig. 54. fore their moments are propor-
tional to P, Q and E. The vectors
representing the moments are perpendicular to AS and to P, Q and E
respectively, thus they form the sides and diagonal of a parallelogram
similar to that of P, Q, E. Therefore couples are compounded by
compounding their moments by the law of addition of vectors.
59 a, 60, 61] KINEMATICAL AND DYNAMICAL DUALISM.
209
6O. Reduction of Groups of Forces. Dualism. Suppose we
have any number of forces applied to various points of a rigid body.
Let one such be P applied at A. At any point
0 apply two equal and opposite forces equal and
parallel to P. One of these P2 forms a couple
with P. The other is equal and parallel to P.
The moment of the couple is perpendicular to
this force.
In this manner the points of application
of all the forces may be brought to 0, where
they can then be compounded into a single
resultant E. For each force thus transferred
there remains a couple, and all the couples
may be compounded into a single one. There-
fore all the forces applied to a rigid body may be replaced by a
single force and a single couple.
We may now state the following dualism existing between
infinitesimal rotations and forces:
Fig. 55.
Infinitesimal rotations are slid-
ing vectors.
Forces applied to a rigid body
are sliding vectors.
When their axes intersect they are compounded by the vector law.
Parallel infinitesimal rotations I Parallel forces
have a resultant parallel and equal
the center of mass of their points
Two equal and opposite parallel
rotations form a rotation -
couple represented by its
moment, a free vector.
Every displacement of a rigid
body may be reduced to a
rotation and a rotation -
couple.
The theory of couples is due
to their algebraic sum, placed at
of application.
Two equal and opposite parallel
forces form a couple, re-
presented by its moment, a
free vector.
Every combination of forces
applied to a rigid body may
be reduced to a force and
force -couple.
to Poinsot.
61. Variation of the Elements of the Reduction. Central
Axis. Null - System. We have seen that any system of slide-
vectors may be reduced to the resultant of a single vector and a
single moment applied at any point whatever. We have now to
examine the variation of the pair of elements, vector E and moment S,
as we vary the point of application 0. E is invariable. As we move 0
along the line of E there is no change since E may be applied at
any point of its axis, and S may be moved parallel to itself. If we
WEBSTER, Dynamics. 14
210 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION,
ft
make the resolution at any other point, 0', the couple to be com-
pounded with S at 0', is perpendicular to E and 00'7 so that if S
has any component parallel to E it
cannot be neutralized by the new
couple. Accordingly in order that the
couple may vanish for any point Of,
the couple S must be perpendicular
to E at all other points. As a change
of 0 introduces only a component
of S perpendicular to E, the com-
ponent parallel to E is unchanged.
Therefore the projection of S on E
is the same for all points 0,
Fig. 56.
Although in general E and S have different directions, we may
find points 0' for which they have the same direction. Let S and E
include the angle & at 0. Resolve S into SQ = S cos #
parallel to JR, and 8^ = 8 sin # perpendicular to .R.
If we take 0' on a line perpendicular to SE at a
distance ^ such that d - E = S sin # in the positive
direction of translation corresponding to a rotation
from E to •$, the component 5t will be neutralized,
and we shall have at 0'} E and S' = $0 in the same
direction. This property holds for all points on the
line of E through 0'. This line is called Poinsot's
central axis.
In order to consider the resolution at any point 0
we may refer it to the central axis. Drop a per-
pendicular from 0 (Fig. 58) on the central axis,
and take this perpendicular for the axis of X, the
central axis for the axis of Z.
Fig. 57.
Then as above
9)
and if xyz are the coordinates of the end of S, we have
-i /^\ a -r) _ E
10) z = /S0, ?/ — *ta> tan ^ = x -« ->
and for any point on the line of /S,
11) — = ^~-> or ## = -^ y,
y E x Eyj
that is the line of 8 lies on a hyperbolic paraboloid.
61]
POINSOT'S CENTRAL AXIS.
211
It is evident that if we slide the whole of Fig. 58 along or turn
it around the central axis nothing is changed, consequently if we
suppose the vector S laid off at
every point of space 0, and con-
sider the assemblage of couples
thus formed, the assemblage re-
mains unchanged if we rotate it
about or slide it along the cen-
tral axis.
Every S is tangent to a cer-
tain helix, or locus of a point
which moves on a circular cylinder
, . J Fig. 58.
in a path making a constant angle
with its generators (Fig. 59). This angle is less as the diameter of the
cylinders is less, so that
E
10)
tan# = # ,
All these helices have however
one constant in common,
namely the distance traversed
parallel to the central axis for
each turn. If dr be the trans-
lation for a rotation do, we
have
xdo
*s:
&
Then
12)
I>
Fig. 59.
is the traverse for each turn,
and is called the pitch of the
helix. Every helix lies on a ruled screw -surface, made by the revolution
of a line perpendicular to the central axis, which slides along it a
distance proportional to the angle of rotation, the pitch of the screw
o
being p = %TC --JJ- The lines of the assemblage of moments have every
direction in space — there are a triple infinity of lines of the system
(one for each point in space), but only a double infiriity of direc-
tions — therefore every plane cutting all these lines has for rjjfcs
points (a double infinity), every possible direction for S. Fpj? Qtye
point only is this perpendicular to the plane. This point ,-ii
14*
212 VI- SYSTEMS OF VECTORS. DISTBJBUT. OF MASS. INSTANT. MOTION.
the focus of the plane. Let the plane cut the central axis in A.
Through A draw a plane perpendicular to the central axis, intersecting
the given plane in AO. As we go along the line AO, S turns
about it, and for one point
has the direction of the
normal to the given plane.
Accordingly to every
point in space there corre-
sponds one plane, and to
every plane one point.
The correspondence was
discovered by Chasles, and
the system of points and
planes was called a Null-
Fig. 60.
System by Mobius.
62. Vector -cross. Besides the reduction to the screw -type
we may reduce the system of vectors to two vectors not lying in
the same plane, without a couple. This reduction may be made in
an infinite number of ways, and the line of one of the vectors may
Fig. 61.
be given. Let AB (Fig. 61) be the given line. At any point 0
on AB let E be the resultant vector, S the resultant couple.
(E and 8 will not in general lie in a plane with AB.) At 0 pass
a plane perpendicular to S, intersecting the plane of E and AB
in OP. Resolve 8 into the pair of vectors OP and CQ so taken
that the resultant of E and OP shall lie in AB. The length of OP
is thus determined, and the distance between its line and that of CQ
is determined by S. Thus the line AB determines the line CQ.
61, 62] NULL-SYSTEM. VECTOR-CROSS. 213
The lengths OB and CQ are determined as soon as the line AB is
given. Two such non- parallel and non-coplanar vectors OB, CQ will
be termed a vector -cross. The crossing will degenerate to intersection
only when S = 0 and to parallelism when R = 0.
As any line may be taken for AB, and as there are a quadruple
infinity of lines in space, there are a quadruple infinity of vector-
crosses. They all possess a property in common, namely, that the
tetrahedron formed by joining the four ends of a vector- cross has a
constant volume. Let OB, CQ (Fig. 61) be the vector- cross, and let
us reverse the preceding resolution. The volume of a tetrahedron is
equal to one -third the product of its altitude by the area of its
base. The area of the base OCQ is one-half the moment of CQ
about 0, or --- S, while the altitude is the projection of OB on the
perpendicular to OCQ, that is, on S. But since BE is parallel to
the plane OCQ, OR has the same projection on S as OB, namely
.Rcos'fr, consequently
v= 4^cos#-4-#=4-^£cos#.
O £4 U
But by 8),
$COS# = SQ,
therefore
13) V=±RSQ.
This theorem is due to Chasles.
Corresponding lines of vector -crosses possess a remarkable relation
to the null-system. Let AB and CQ (Fig. 62) be the two lines of
the vector- cross. Through CQ pass any
plane, cutting AB in 0. The moment
of CQ is perpendicular to the plane OCQ,
and the other vector has no moment
about 0, since it passes through it. Accord-
ingly 0 is the focus of the plane OCQ.
Thus, if a plane turns about a line, its
focus traverses another line, and these
two conjugate lines are lines of a vector-
cross.
We have here shown the intermediate
nature of a line between a point and a
plane, in the dual role as generated by the motion of a point and
by the rotation of a plane. In the first relation the line is spoken
of as a ray, in the second as an axis.
If two conjugate lines are at right angles, pass a plane through
one, AB, perpendicular to the other, CD (Fig. 63). By the preceding
214 VI. SYSTEMS OF VECTORS. DISTEIBUT. OF MASS. INSTANT. MOTION.
theorem, the point of intersection of the plane with CD is the focus
of the plane. Resolving at any point P in AB, the moment of OD,
being perpendicular to OD
E and OP, lies in the plane
OAR
That line in a plane which
has the property that for all
its points the resultant
moment lies in the plane is
called the characteristic of
the plane, or of its focus.
Its distance OX = d from the
focus is such that1)
14) dEsmfr = S.
The line OX, of length
S
Rsinfr'
d
S
is perpendicular to the plane
of E anc( S, and drawn toward the side corresponding to the motion of
a right-handed screw when rotated in the direction from E to S. If
we should go from 0 in the direction OX a distance d' = — ^ — we
should reach the central axis, and
15) dd' = W'
63. Complex of Double -lines. If a plane 1 pass through
the pole of a plane 2, then the plane 2 passes through the pole of
the plane 1. Let P (Fig. 64)
be the pole of the plane 1, and
let PO be any line in 1 through P.
The moment of E about 0 is
perpendicular to PO, and so
is S, hence so is their resultant.
Thus the moment at 0 is per-
pendicular to OP, and the polar
plane of 0 contains the line OP,
that is, if 0, the pole of 2 lies
in 1, then P, the pole of 1 lies
in 2.
In this case the two poles lie in the line of intersection
of the planes, and we see that if a plane turns about a line through
its pole, its pole traverses that line. Such a double line is conjugate
Fig. 64.
1) For the component in AS, E sin #, has the moment S about 0.
62, 63] COMPLEX. FLICKER'S COORDINATES. 215
to itself. The necessary and sufficient condition that a line is self-
conjugate is that the pole (focus) of a plane through the line falls
in the line. For then as the plane rotates about the line as an axis,
the focus describes the line as a ray. Hence the double lines lying
in a particular plane all pass through the pole of that plane , and
conversely, all the double lines passing through a point lie in the
polar plane of the point. Such a system of lines is called by Pliicker
a line complex of the first degree. There are in all a double infinity
of lines passing through any point in space, but of these only a
single infinity belong to the complex. Therefore lines belonging to
the complex have one less degrees of freedom than lines in general,
or a complex contains a triple infinity of lines. A complex may be
represented analytically by a single relation between the four para-
meters determining a line. If we mark off on a line any length B,
and give its projections on a set of rectangular axes X, Y, Z, and
the projections L, M, N of its moment about an origin 0, the line
is completely determined. For its direction is given and giving the
moment S = T/L2-\- M2 -\- N2 gives the plane through 0 containing R,
and the distance from the line, if the length of R is given, but this
is given by R = ]/X2 + Y2 + Z2.
As the determination of the line is independent of the length
of JR, the ratios of the six quantities determine the line. But these
five ratios are not independent, for since by § 5, 12),
16)
we have the identical relation,
17) LX + MY+ NZ=0,
expressing the fundamental property that the moment of a vector is
perpendicular to it. The coordinates LMNXYZ are known as
Pliicker's line -coordinates.
Thus there remain four independent quantities to determine a
line. A relation between these denotes a complex, and in particular
a linear relation,
18) aX + IY + cZ+dL + eM+fN= 0,
denotes a complex of the first degree.
Since the double lines of the null- system are the loci of points
which are the poles of planes containing the double -lines, at every
point of a double -line the resultant moment is perpendicular to it,
216 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
or double lines are lines of no moment. In the kinematical applica-
tion, points on a double -line experience no translation along it.
If a double -line cuts one of a pair of conjugate lines, it cuts
the other. Let PQ be a double -line cutting the line AS. Then
the pole of the plane BPQ lies in the line conjugate to AS. But
since PQ is a double-line, the pole of BPQ lies on PQ. Hence
PQ cuts the conjugate to AB. Conversely, every line cutting two
conjugates is a double -line.
The complex of double -lines is symmetrical with respect to the
central axis. Let AB (Fig. 65) be a line of the complex, and let OX
be the common perpendicular to it and
the central axis. Now AB is perpen-
dicular to the moment S at X, but S
is perpendicular to OX, and the distance
o
OX is d = -jr tan #-. If
* = sin 2a*>
cos cc sin2
= ^ cos2 «2 + py sin2
as six equations to determine px, py, sl} s2, aly cc2. We have by
elimination
£2 — 2*1 = — . (sin 2a2 — sin2aj)
or using the first two equations,
27) A = -J^S cos (oj + ai) sin y,
P —px
28)
Pv—px .
= -^ — - sin (cc2 -f KI) cos (K2 — «j
~~ Pi = P* (cos2 a2 — cos2 aj) + py (sin2 a2 — sin
= (Px —Py) (cos2 a2 — cos2 ^)
= (Px -Py} sin (ag + ai) sin («2 - ^);
Pz-Pi = (Px - Py) sin (ag + «0 sin y,
+ (Px -Py) (cos2 cc2 - sin2 aj
= Pz+Py "f (P* -jPy) COS (^ -f C^2) COS y.
From 27) and 29) we obtain
(_pj -jpj2 = (py -px}2 sm2y;
31) ^-^
smy
From 27) and 30),
From 29) and 31),
- **~Pl
COS (^ + Oj) = :?=
32) tan (^ -
220 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
From 32) with a2 — a± = y,
Since the cylindroid is thus determined by 31) and 33), a twist
about PI can be resolved into a twist about px and one about py.
A twist about p2 can be likewise resolved. The two components
about px add together, so do those about py, and since the resultant
of any twists about px and py lies on the cylindroid , the resultant
of p1 and p2 does. Its direction can be found, since the amplitudes E
of the two twists about p^p2 compound by the parallelogram law,
hence the angle made by the resultant with the axes is known. The
pitch is then found from the pitch -conic.
65. Work of Wrench in Producing a Twist. Let us find
an expression for the work done during a twist of amplitude Rk
about a screw of pitch pk by a wrench of intensity Ef about another
screw of pitch pf. We already know the work done by a force in
a translation, namely, it is equal to the product of the magnitudes
by the cosine of the included angle. If the force is Rf and the
translation (rotation -couple) is $#, we have
W = EfSkcos(EfSk).
Notice that the vector of one system is multiplied by the vector-
couple in the other.
We can find the work done by the force -couple in a rotation
about its axis. Apply the couple so that one of its members P
passes through the axis of rotation. In a rotation this member does
no work, for its point of application is at rest, while that of the
other member Q moves in a rotation a distance da, where d is the
arm of the couple. Accordingly the work is W=Pdc3 which is
equal to the product of the twist by the moment of the couple.
Here again we multiply the vector of one system by the vector-
couple of the other.
If the axis of rotation is perpendicular to the axis of the couple,
the motion is perpendicular to the force, and no work is done. Hence
we must take the resolved part of the couple on the vector, as before.
We can now find the work of a wrench during a twist. The
work of the force in the displacement Sk is jR/^cosa, a being the
angle between the two screws. The work of the couple Sf = -^— Ef
in the rotation Ek is
Pf
SfEk cos a = ~ EfEk cos cc.
65, 66] RECIPROCAL SCREWS.
But when Ef is changed to the origin of Rk it gives rise to a moment
perpendicular to Ef equal to Efd, d being the perpendicular distance
between the screws. This moment therefore makes with Ek the
angle a -j- -g-> and the work done by it in the rotation Ek is
dEfEk cos a -f y = — dEfEk sin «.
Thus the whole work is
34) W = EfEk ( cos a - d sin a .
It is symmetrical with respect to both screws, hence the wrench and
twist might have been interchanged.
The geometrical quantity in parentheses is called the virtual
coefficient of the two screws, and if it vanishes no work is done,
that is, a body free to twist only about a particular screw is in
equilibrium under a wrench about another screw if the virtual coef-
ficient of the two screws is zero. The two screws are then said to
be reciprocal.
66. Analytical Representation. Line Coordinates. In
Pliicker's line coordinates referred to any origin, since each component
of vector does work on the corresponding component of couple in
the other system,
35) W = XfLk + YfMk -f ZfNk + LfXk + MfYk + NfZk.
If a screw is reciprocal to two screws on a cylindroid, it is
evidently reciprocal to all the screws on it.
For two screws to be reciprocal, the condition is,
§36) X,L2 + Y,M, + Z,N2 + L,X2 + M,Y2 + N,Z, = 0.
If the coordinates of one of the screws be constant, while those of
the other be variable, this is the equation 18) of a complex of the
first degree, so that all the screws reciprocal to a given screw form
such a complex.
Since between the six coordinates X1Y1Z1L1M1N1 there is
always the identical relation
X, A +Y1M1 + Z1N1 = 0,
we may always make them satisfy five equations like the above,
that is, we may always find a screw reciprocal to five arbitrarily
given screws.
Suppose the coordinates of the system of vectors for an origin 0
are XYZLMN, being the projections of E and 8 at 0. Let
222 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
XYZ L'M'N' denote the same for a point 0' whose coordinates
Then
.37) M=M' +
N=N' + xY-yX.
In order that the point O1 may lie on the central axis, the direction
of resultant and couple must coincide , or
*L = M- = *[L
X~ Y ~ Z '
hence the equations of the central axis in Cartesian coordinates are
L-yZ-\-zY_ M-zX + xZ N-xY+yX
~^~ ~Y~ ~W~
The equation of the focal or polar plane to a point x' yr zr is,
since it is perpendicular to L'M'N',
39) (x-x')L' + (y-y')M' + (2-z')N' = 0
and inserting the values of L'M'N',
or, more symmetrically arranged,
40) L(x-x') + M(y-y') + JV (*-*')
+ Y(xe' - 0x') + Z(yx' - xy') = 0.
This equation is symmetrical with respect to xyg, x'y'z', hence if
x' y' z' is fixed, xyg is on its polar plane, or if xyz is considered
fixed, x' y' &' is on its polar plane, showing the reciprocal relation
of pole and polar.
If the vector system is to reduce to a single vector, the resultant
and couple at any point must be perpendicular, or
41) LX + MY+NZ = 0.
We must have in general, at any point, $cos# = $0 that is,
42)
and the pitch p is given by
4QN P 3, LX+MY+NZ
2^t == ~R = ~"X8+ r*-f Z2
The volume of the tetrahedron on a vector -cross is
44) I ES, = \(LX+ MY+ NZ),
and this, like the last expression, is independent of the choice of
origin or axes, that is, is an invariant.
66] REPRESENTATION BY LINE -COORDINATES. 223
Suppose that the two members of a vector -cross have Pliicker's
coordinates
X^Z^M^ and X2Y2Z2L2M2N2
with the identical relations,
Their resultant has components
and the volume of the tetrahedron is one sixth of
LX + MY + NZ =
which in virtue of the two identities is
45) L,X2 + M,Y2 + N,Z2 4 L2X, + M2Y, + N&.
If any two lines are given by their Pliicker's coordinates, the
condition that they shall intersect is that the above expression shall
vanish.
We may now find the equation of the complex of double -lines.
We have seen that every line meeting two conjugate lines is a
double -line. Let the coordinates of the two conjugate lines be
X1 . . . Nlf X2 . . . N2, satisfying the conditions
46)
^ Z,,
where Xf^Y^Z^L^M^N^ define the vector -system. Let the coordinates
of a double -line be XYZLMN. The condition that it meets the
line X^Z^M^ is
LtX + MJT+ N,Z -f X,L + Y, M + &N= 0,
and that it meets X2Y2Z2L2M2N2)
L2X+M2Y+ N2Z+ X2L + Y2M + Z2N = 0.
Adding these equations, and using the conditions 46) we obtain,
47) L0X+M0Y+Nl>Z+Xi>L +
as the equation of the complex, that is, any linear relation in
Pliicker's coordinates represents a linear complex, as stated in § 63.
It is to be noticed that the equation 47) does not signify that
the line XYZLMN cuts the line XoroZ0L0J!f0JV"0 unless the latter
224 VI. SYSTEMS OF VECTORS. DISTftlBUT. OF MASS. INSTANT. MOTION.
are the coordinates of a line (not of a general system of vectors),
that is fulfill the relation
If they do, then every line of the complex cuts the line X0Y0,ZOJL0Jf0./V0,
and the equation may be considered the equation in Pliicker's co-
ordinates of the line XQYQZQL0MoNQ (see Clebsch, Geometric, Yol. II,
p. 51). For further information on this subject, the reader may
consult, Ball, Theory of
67. Momentum Screw. Dynamics. The previous sections
have shown how to combine systems of vectors having different
points of application, provided they are unchanged if slid along their
lines of direction. As one particular system to which the operation
is applicable we have had the various rotation - velocities of
a rigid body, as another, sets of forces applied to a rigid body.
That these vectors are susceptible of such treatment may be considered
as due to properties of a rigid body, rather than of the vectors
themselves. We have however previously dealt with two other sorts
of vectors which may be dealt with in similar fashion, on account
of their physical nature, and independently of the nature of the
bodies in which their points of application lie. By means of these
properties we are able to connect the kinematical aspect of a rigid
body, as expressed by its instantaneous screw motion, with its
dynamical aspect, as expressed by an applied wrench about another
screw.
If for each point of the system we consider the momentum,
whose six coordinates (one being redundant), in the sense of § 66 are,
mvx, mvy, mvz, m(yvz — 2Vy), m(zvx — xvz), m(xvy — yvx),
and form the general resultant, we obtain a system whose co-
ordinates are
Mx = Zmvx, Hx = Zm (yvz — zvy},
48) My = 2mvy, Hy = 2m (zvx — xvz},
Mz = 2mvZJ Hz = Urn (xvy — yvx),
which represent the momentum of the system, the three projections
Mx, My, MS, being more particularly characterized as the linear
momentum, the others Hx, Hy, Hz, as the angular momentum or
moment of momentum with respect to the origin.
We have now by the general principles of dynamics, as shown
in § 32, 45), § 33, 61), the fact that the time -derivatives of these
six components of momentum are equal to the corresponding com-
ponents of the resultant wrench,
66, 67, 68] DYNAMICS. IMPULSIVE WRENCH. 225
applied to the system. That is,
dM.. dM
49)
dH dH
Integrating these equations with respect to the time,
we may, in the sense of § 27, call the momentum the impulsive
^vrench of the system. Physically, then, the momentum that a system
possesses at any instant is equal to the impulsive wrench necessary
to suddenly communicate to it when at rest the velocity -system that
it actually possesses. As a prelude to the dynamics of a rigid body
we must accordingly study the properties of the momentum or
impulsive wrench of a body possessing a given instantaneous twist-
velocity.
All the systems of vectors in question may be reduced to the
screw type, and their respective screws are in general all different.
Thus we may speak of the instantaneous velocity -screw and instan-
taneous axis, the momentum screw, and the force -screw. As the
body moves, all these screws change both their pitch and position
in the body, describing ruled surfaces both in the body and in space.
The integration of the differential equations of motion 49) will enable
us to find these surfaces. The kinematical description of the motion
will be complete if we know the two ruled surfaces described in
space and in the body by the instantaneous axis, together with such
data as will give their mutual relations at each instant of time.
68. Momentum of Rigid Body. The properties of the
momentum of a rigid body are conveniently investigated by the
consideration of the velocity -system as an instantaneous screw -motion.
Let F be the velocity of translation, and co of rotation. Then every
particle of mass m has one component of momentum parallel to the
axis of the instantaneous twist (which we will take for Z-axis),
equal to mvz = mV and the resultant for all is
51) M2 = ZmV=VZm = MV,
WEBSTER, Dynamics. 15
226 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
where M is the total mass of the body. By the construction of
§§ 57, 59 the resultant of parallel vectors P and Q is applied at the
center of mass of masses proportional to P and Q placed at their
points of application. Consequently the various elements being pro-
portional to the masses m,
this component of the
momentum is applied at
the center of mass of the
body.
There remains the
component of momentum
perpendicular to the instan-
taneous axes. Let OZ
(Fig. 68) be the instan-
taneous axis, and let r be
the perpendicular distance
from it of any point ^Pf
and let the angle made
by r with the X-axis be #.
Now P is moving parallel
Fig. 68. to the XY- plane with the
velocity v = r& perpendic-
ular to r, so that the projections of this velocity are
vx = — v sin & = — G)r sin # = —
v
vcos&
or cos -9-
ox.
Thence we obtain the components of momentum
Mx = — 2 may = — oZmy = — May,
52) K/r _
where x, y are the coordinates of the center of mass. The resultant
momentum is accordingly equal and parallel to the momentum that
the body would have if concentrated at the center of mass, but its
point of application is different, for the components Mx, My are not
applied at the center of mass, inasmuch as their elements are pro-
portional, not to m but to my and mx. The magnitude of the
resultant momentum being given by Mx, My, M~, we may find its
axis by obtaining its three remaining coordinates, representing the
angular momentum. We have
53) Hy =
Hz =
MVy —
= — MVx
= &2m (x2 -f- y2) =
68] MOMENTS AND PRODUCTS OF INERTIA. 227
Of these the terms in V are the moments of the vector M V
in the direction of the Z-axis applied at the center of mass, while
the terms in o are applied elsewhere. The equations of the central
axis of momentum are, by § 66, 38), x' y! z' being the running
coordinates,
Hx
or inserting the values,
•> r\ — (o Zmxz — y' MV -f z' Miax
- M.V~x — a Emyz + z' Mo>y -f x' MV
M&HC
_ co Zm (x* -f y2) — x' Max — y' May
MV
This does not pass through the center of mass unless, putting
x! = x, y' = y, z' =~z,
*K\ — aZmxz -f Mazx _ — oEmyz -f M&yz
— M.(oy
« _ co Zm (xz -f 2/2) — M a) (x
MV
We see that the resultant momentum involves the various sums
ZmXj Zmy, Zmxz, 2myz, Hmr2,
the axis of Z being the instantaneous axis. These sums are constants
for the rigid body, depending on the distribution of mass in it. The
first two represent the mass of the body multiplied by the coordinates
of the center of mass. The last represents the sum of the mass of
each particle multiplied by the square of its distance from the Z-axis,
and is what has been called the moment of inertia of the body with
respect to that axis. We are thus led to consider the sums
A = Zm (y2 4- A B = 2m (z2 -f #2), C = Zm (x2 + y2),
D = Zmyz, E = Zmzx, F = Hmxy.
Of these the last three, D, E, F, are termed the products of inertia
with respect to the respective pairs of axes.
In the case of a continuous distribution of mass, we must divide
the body up into infinitesimal elements of volume dt, and if the
density is 0, the element of mass is dm = gdt and the six sums
become the definite integrals
15
228 VI. SYSTEMS OP VECTORS. DISTBIBUT. OF MASS. INSTANT. MOTION.
The determination of these quantities is then, like that of centers of
mass, a subject belonging to the integral calculus.
The six constants A, B, C, D, E, F together with the mass M
and coordinates x, ?7, 0", of the center of mass, completely characterize
the body for dynamical purposes, since when we know their values
and the instantaneous twist, the momentum or impulsive wrench is
completely given. The body may therefore be replaced by any other
having the same mass, center of mass, and moments and products
of inertia, and the new body will, when acted upon by the same
forces, describe the same motion.
69. Centrifugal Forces. As the body moves, its different
parts exercise forces of inertia upon each other, so that there is a
resultant tending to change the instantaneous screw in the body.
Let us suppose the translation to vanish, and examine the kinetic
reactions developed by the rotation, or the centrifugal forces. The
instantaneous axis being again taken as the axis of Z, a particle P
experiences the centripetal acceleration — = ro2 towards the axis, and
the centrifugal force is Jtc = mrs? (see p. 119) directed along the
radius r from the axis OZ, and having the projections
Zc= 0.
For the moment of the centrifugal force we have
Lc = yZc — zYc = — my #«2,
58) Mc = 2Xc—xZc = rnxsG?)
Nc = xYc-yXc = 0 ,
so that the coordinates of the resultant centrifugal force and couple are
Xc= tfZmx = a* MX,
59) c
Lc = — &
Nc= 0 .
Thus the centrifugal force is equal and parallel to that of a
mass placed at the center of mass, and moving as the latter point
does. It vanishes when the center of mass lies in the axis. The
system of centrifugal forces is however, as in the case of the
68, 69, 70, 71] CENTRIFUGAL FORCES. 229
momentum, not to be replaced by a single force placed at the center
of mass, for the couple is not equal to what its value would be in
that case, unless — = -|-- If the center of mass lies on the axis,
although the centrifugal force Rc vanishes, the centrifugal couple 8C
does not, unless D = E = 0.
The centrifugal forces then in general tend to change the instan-
taneous twist, unless the axis of the latter passes through the center
of mass, and for it D = E = 0. Such axes are called principal axes
of inertia of the body at the center of mass, and are characterized
by the property that if the body be moving with an instantaneous
twist about such an axis, it will remain twisting about it, unless
acted on by external forces. In order to examine the effect of the
distribution of mass of the body, we are led to interrupt the con-
sideration of dynamics in order to consider the purely geometrical
relations among moments and products of inertia.
7O. Moments of Inertia. Parallel Axes. Consider the
moments of inertia of a body about two parallel axes. Let the
perpendicular distances from a point P
on the two axes be p± and p2 and let
the distance apart of the axes be d.
Let A and B (Fig. 69) be the inter-
sections of the axes with the plane of
p± and p2. If we take AS for the
X-axis, A for origin, we have rig> 69.
P^ = Pi* + d2- ^d cos (p^x),
60) Zmp = Zmp^ + Md2 - 2dZmp1 cos (p±x)
The last term is equal to — 2dMx and vanishes if the axis 1 passes
through the center of mass. Consequently the moment of inertia
about any axis is equal to the moment of inertia about a parallel-
axis through the center of mass plus the moment of inertia of a
particle of mass 'equal to that of the body placed at the center of
mass, about the original axis. Consequently of all moments of inertia
about parallel axes, that about an axis through the center of mass
is the least. In virtue of this theorem the study of moments of
inertia is reduced to the study of moments of inertia about axes in
different directions passing through the same point.
71. Moments of Inertia at a Point. Ellipsoid of Inertia.
Consider now moments of inertia about different axes all passing
through the same point 0. Let a, /?, y be the direction cosines of
230 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
any axis. Let p be the perpendicular distance of a point P from
the axis, r its distance from 0, and q the distance from 0 of the
foot of the perpendicular. Now since q is the projection of r on
the axis,
61) q = ax + fty + yz,
and we have
tf = r* - = x* + f + z* - (ax
62) =^(i_^) + 2/2(1_^) + ^2(
- 2(fiyyz + yazx + afixy).
Now since we have
- « = + y, -/ = r+«,
and replacing in 62),
4-
-f
-f
Thus the moment of inertia K about any axis whose direction cosines
are a, /3, y, is given by
64) K=Aa* + Bp+Cf-2Dfly-ZEra-2Fap = F(a,p,y),
as a homogeneous quadratic function of the direction cosines of
the axis.
The sum of products of the mass of each particle multiplied by
the square of its distance from a given plane is called the moment
of inertia of the system with respect to the plane. Although it has
no physical significance it will be convenient to consider it. For a
plane normal to the preceding axis we have
65) Q = Zmq2 = a?Zmx* + (P2my* + fZmz2
+ SfiyZmyz -f 2yaZmzx + 2a(lZmxy,
and if we put
A' = Zmx2, B' = 2my*, C' = 2mz*,
we have
66) Q = A!a2 + B'p*+C!f + 2I>pr+2Era + 2Fap = F'(a,p,>y*).
The six quantities, A, B, C, A', B\ Cr, being sums of squares, are all
positive. We have evidently
67)
71] ELLIPSOID OF INERTIA.
so that the sum of any two of the moments A, B, C is greater than
the third.
If we lay off on the axis a length p and call the coordinates of
the point P so determined |, y, g we have
If we now make the length of OP vary in such a manner that
Q2 Q = 1, we obtain for the coordinates of P the equation
69) *"(!, ,, g) -4'{» + JBV + '£2 + 2.0^ + 2.^ + 27^ = 1,
or P lies on a central quadric surface. Since p = — = is always real,
V V
this is an ellipsoid. It possesses the property that the moment Q
with respect to any plane through its center is inversely proportional
to the square of a radius vector perpendicular to it. It will be
termed the fundamental ellipsoid of inertia at the point 0. It was
discovered by Binet.
In a similar manner the moments of inertia about the various
axes are inversely proportional to the square of the radii vectores in
their direction of another ellipsoid
70) F(t, v, Q = A? + Btf + C? - 2Drt - 2E& - 2D^ = 1.
This is known as Poinsot's ellipsoid of inertia at the point 0.
Since a central quadric always has three principal axes perpen-
dicular to each other (see Note IV), we find that there are at any
point in a body three mutually perpendicular directions, namely those
of the axes of the two ellipsoids of inertia, characterized by the
property that for them the products of inertia D, E, F, are equal to
zero. These are termed the principal axes of inertia of the body at
the point in question. They have, as shown in § 69, the property
that if the body be rotating about one of them the centrifugal couple
vanishes, so that if the center of mass lies on the axis the body
remains rotating about the same axis, unless acted on by external
forces.
The moments A, B, C about these axes are called principal
moments of inertia.
It is important to notice that as we pass along a line which is
a principal axis at one of its points, the directions of the axes
of the ellipsoids at successive points are not the same, so that in
general a line is a principal axis of inertia at only one of its points.
We are thus led to study the relative directions of the principal
axes at different points of the body.
232 VI. SYSTEMS OF VECTORS. DISTBIBUT. OF MASS. INSTANT. MOTION.
72. Ellipsoid of Cry ration. The moment of inertia about any
axis may be considered equal to that of a particle whose mass is
that of the body placed at a distance Js from the axis, such that
K = MJc2. It is called the radius of gyration for this axis. The
radii of gyration about the principal axes of inertia at any point are
called the principal radii of gyration for that point. If we call their
lengths a,l),c we have
and 70) becomes
Another ellipsoid besides Poinsot's, which referred to its axes is
72) F(x, y, e) = Ax2 + By2 + Cz2=l
is sometimes convenient. If at any point x, y} z on Poinsot's ellipsoid
we draw the tangent plane, and from the center let fall a perpen-
dicular upon it, its length p will be the projection of the radius
vector r on a line parallel to the normal,
73) p = x cos (nx) + y cos (ny) -f z cos (nz).
But since
Ax
rjA\
this gives for the ellipsoid
dF . dF . dF
Thus the direction cosines of p are, by 74),
a' = cos (nx) = Apx = Apr a,
75) j8f = cos (ny) = Spy = Bprfi,
y' = cos (nz) = Cpz = Cpry.
If on the perpendicular we mark off a point P' at a distance
-R2
OP' = r' = — 9 and call its coordinates x',y\z\ we have
72]
ELLIPSOID OF GYRATION.
' = r'a' = Apr'x = B*Ax,
233
76) y'
from which we obtain
77) x =
and by 72)
78)
»'
BR*
Of"
^f
Accordingly the locus of P' is an ellipsoid, whose axes are
inversely proportional to those of the original ellipsoid. It is called
the inverse ellipsoid. If we take
irp we have
M
79)
_L — +
a2 1 62 ~~ c2
1
and the semi -axes of the inverse
ellipsoid are equal to the principal
radii of gyration a, fr, c.
Since the two ellipsoids have the
directions of their principal axes coin-
cident (namely the directions in which p
and r coincide), the relations are
evidently reciprocal, and OP is per-
pendicular to the tangent plane at P'.
Let the length of the perpendicular in
this direction be p'. Then since the triangles OPQ, OP' Q' (Fig. 70)
are similar,
p p' '
Since the moment of inertia about OP is
we have
Fig. 70.
and the property of the inverse ellipsoid is that the radius of gyration
about any line is equal to the part intercepted by a plane perpen-
dicular to it tangent to the inverse ellipsoid. The inverse ellipsoid
is accordingly called the ellipsoid of gyration.
It is evident that the direct ellipsoid more nearly resembles the
given body in shape than the inverse ellipsoid, for if the body is
spread out much about any particular axis the inertia and radius of
234 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
gyration about that axis are large, so that the inverse ellipsoid has
a large dimension in the direction of that axis, while the direct
ellipsoid, like the body, has a small one.
73. Ellipsoidal Coordinates. The equation of a central
quadric referred to its axes may be written,
'82) £L' + JL'+£! = i,
a, ^ az ^ as
where alt a%, a3, may be positive or negative. If they are all negative,
the surface is imaginary, for the equation is not satisfied by any
real values of x,y, 2.
1°. Suppose one is negative, say
% = - c2,
while
a^ = a2, a.2 = b2.
Let
a > I > c.
The equation now is
/y.2 ,,.2 ,.,2
fL i y ___ — — \
a? + 52 c2 ~
The surface is cut by the XY- plane in the ellipse
whose semi -axes are a and 5, and whose foci are at distances from
the center
/a2 — b2 = /% — a2
on the X-axis.
The section by the ZX~ plane is the hyperbola
with semi- axes a, c, and foci at distances "j/a2 -}- c2 = y^ — aB on
the X-axis. The section by the YZ- plane is the hyperbola,
with semi -axes fc, c, and foci at distances "J/62 -}- c2 = j/a2 — a3 on
. the F-axis. The surface is an hyperboloid of one sheet.
2°. Let two of the constants a19 a2, a3, be negative, say
The equation is
72, 73] ELLIPSOIDAL COORDINATES. 235
The sections by the coordinate planes and their focal distances are
XY - = 1 Hyperbola, i/o^+T2 = V^~^ on X-axis,
^ - - ~ = 1 Hyperbola, ]/a2T72 = Va^a, on X-axis,
* -f = - 1 Imaginary Ellipse, /-(62_c2) = Va^a*.
The surface is an hyperboloid of two sheets.
3°. If al9 a2, as are all positive, the sections are all ellipses, and
the surface is an ellipsoid. In all three cases, the squares of the
focal distances are the differences of the constants %, &2, a3. Con-
sequently if we add to the three the same number, we get a surface
whose principal sections have the same foci as before, or a surface
confocal with the original. Accordingly
^2 _L y* + g2 _ i
-
represents a quadric confocal with the ellipsoid
tf + V + ^ = 1?
for any real value of Q.
If a > 6 > c and p > — c2, the surface is an ellipsoid. If
— c2 > Q > — &2, the surface is an hyperboloid of one sheet, and if
- Z>2 > Q > — a?j an hyperboloid of two sheets. If Q < — a2, the
surface is imaginary.
Suppose we attempt to pass through a given point x, y, s, a
quadric confocal with the ellipsoid
Its equation is 83), where the parameter Q is to be determined.
Clearing of fractions, the equation is
84) ft?) EH (o« + 9) (6* + 9) (o2 + 9) - a? (V + 9) (c2 +
a cubic in p. But this is easily shown to have three real roots.
Putting successively p equal to oo, — c2, — V, — a2 and observing signs
of f(9),
P= oo, f(9) = °°
236 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
Call the roots in order of magnitude A, ^, v. The changes of
sign above show that I lies in the interval A > — c2 necessary in
order that the surface shall be an ellipsoid, ^ in the interval
- c2 > ^ > — fr2 that it may be an hyperboloid of one sheet, and v
in the interval — fr2 > v > — a? that it may be an hyperboloid of
two sheets. There pass therefore through every point in space one
surface of each of the three kinds. If we call
85) F(l, x, y, ,) = -^ + -^ + -^ - 1,
the equation F=Q defines A as a function of x,y,8. The normal
to the surface A (xyz) = const, has direction cosines proportional to
til dl dl
o — * o — y ^^'
ex cy cz
Now since identically _F = 0, differentiating totally,
3F , dF , dF , . 0.F , .
and we have
dF
for the required partial derivative of A with respect to #, when
and 0 are constant.
Therefore
31 2x I i x* * z* 2 a;
Similarly
86)
dz
The sum of the squares of the derivatives being called /^2, we have
4
Now the direction cosines of the normal to the surface A = const, are
cos (n*x) = T- o— = ±
88) cos (n^y) = +
cos
73, 74] COORDINATE SURFACES ORTHOGONAL. 237
Similarly for the normals to the surface p = const,,
cos
89) cos (n^y) =
cos (nuz) =
The angle between the normals to A and /A is given by
f #2 w2
^ ^ \(a2~F ^) l^2-}- ^) (^24~ ^) (^2~F f1-)
i__
Now by subtracting from the equation
_x<
the equation
,2 W2
-f
we obtain
/ ^1 «2 J_ 1 «2J_ „ f I y \ 13 I 1 7,2 I ., I I
or
2/2
f,v,2 »»2 «2 \
(a« + ao(a* + fO + (&2 + ^)(&2 + ^) + (C2 + ^)(c2 + ")J ^
Accordingly, unless h = ii, cos (n^n/^) = 0, and the two normals .are
at right angles. Similarly for the other pairs of surfaces. Accord-
ingly the three surfaces of the confocal system passing through any
point cut each other at right angles.
If we give the values of A, ^, v we determine completely the
ellipsoid and two hyperboloids, and hence the point of intersection
x, y, 8. To be sure there are the seven symmetrical points in the
other quadrants which have the same values of A, ft, v, but if we
specify which quadrant is to be considered this will cause no
ambiguity. Thus the point is specified by the three quantities >L, ^, v,
which are called the ellipsoidal or elliptic coordinates of the point.
74. Axes of Inertia at Various Points. Let K = MW be
the moment of inertia about an axis whose direction cosines are
«, ft y, at a point 0 whose coordinates with respect to the principal
axes at the center of mass G- are xyz. Let p be the distance of the
axis at 0 from a parallel axis through 6r, and q the distance of the
238 VI. SYSTEMS OF VECTOES. DISTBJBUT. OF MASS. INSTANT. MOTION.
foot of the perpendicular from 6r. Then by the two theorems of
§ 70 and § 71,
K = Ace2 + BP* + Cf +
fc2 = a2a2 + 62/32 + c2y2 -f p\
Now
tf = r2 - (f = r2 - (ax + /3y + r*)2,
94) &2 = ^2 + £202 + c2;;2 _j_ ^2 _ q2
In order to find the principal axes at 0 we must make this a
maximum or minimum with respect to a, ft y subject to the condition,
a2 + /32 -f r2 = I-
Multiplying this by a constant tf, subtracting from 94), and diffe-
rentiating
95) {^2-<5(
Multiplying these equations respectively by «, ft y and adding,
a2 a2 + tfp + cV - g (aa; + fty + ye) - (3 = 0,
fc2 _ r2 _ 6 = 0<
Thus tf is determined as
97) 6 = W- r2.
Inserting this value in 95) we have
(a2 + r2 - F) a = qx,
98) (&2 + r2 - F) /3 = 2y,
(c2 +ra-fc8)y =g^.
Multiplying these equations respectively by
x y e
and adding, we get, since q divides out,
" " 4- z* - 1
-- -
If we now put r2 — &2 = ^), this is the same cubic as 83) to
determine 9, and gives three real roots for &2,
7. 2 _ 0.2 1
/fcj — T ~ A ,
74] DISTRIBUTION OF PRINCIPAL AXES. 239
The direction cosines are then given, according to 95) and 98), by
f («2 + r*- V) - f (V + r* - V) = f (c2 + r* - kf),
100) S(^ + ^-V) = (62 + ^-V) = f(c2 + »-2-V),
that is
etc.
Hence the principal axes of inertia at any point 0 are normal
to the three surfaces through 0 confocal with the ellipsoid of gyration
at the center of mass. This theorem is due to Binet.
Since A, > /z, > v, the least moment of inertia is about the normal
to the ellipsoid, the greatest about the two -sheeted hyperboloid, and
the mean about the normal to the one- sheeted hyperboloid.
We have
It* + ^ + fa* = 3r2 - (I + 11 + *,).
But the sum of the three roots is the negative of the coefficient
of 02 in the cubic 83),
I -f 11 + v = x* + f ' + z2 - (a2 + V + c2),
101) ^2 + ^22 + ^32 = 2r2 + a2 + 62 + c2.
Thus the sum of the principal moments of inertia is the same for
all points lying at equal distances from the center of mass.
It is now easy to see that any given line is a principal axis for
only one of its points, unless it passes through the center of mass,
when it is such for all of its points. It is also evident that not
every line in space can be a principal axis.
If the central ellipsoid of gyration is a sphere, all the ellipsoids
of the confocal system are spheres, and all the hyperboloids cones.
Every ellipsoid of inertia is a prolate ellipsoid of revolution, with
its axis passing through the center of mass.
If the central ellipsoid has two equal axes, the ellipsoids of
inertia for points on the axis of revolution are also of revolution.
If the distance of a point on this line from the center of mass is d>
and the moment of inertia about it is M Jc^
240 VI. SYSTEMS OF VECTORS. DISTEIBUT. OF MASS. INSTANT. MOTION.
If & < & there are two points for which the ellipsoids of inertia are
spheres, namely where d = + ]/a2 — fr2. This is the only case, except
the above, where there are spheres.
If we look for ellipsoids of revolution in the general case when
«, &, c are unequal, we must distinguish between prolate and oblate
ellipsoids of gyration.
1°. Prolate. The two equal radii of gyration are the two smaller
\ and \ . For these to be equal, we must have k = p. But as A
and ^ are separated by — c2, if they are equal they must be equal
to — c2. In this case the axis of the ellipsoid and one -sheeted
hyperboloid are both zero, and the ellipsoid becomes the elliptical
disk with axes }/a2 — c2, }/&2 — c2, forming part of the XF-plane,
and the hyperboloid all the rest of the XT- plane. Points lying on
both surfaces lie on the ellipse whose axes are "/a2 — c2, "J/fr2 — c2,
which passes through the four foci of the system lying on the X-
and F-axes, and is accordingly called the focal ellipse of the confocal
system. (We saw by 92] that if A = fi the two surfaces were not
necessarily orthogonal.) All points lying on this ellipse have prolate
ellipsoids of gyration, the axes of rotation lying in the plane of the
ellipse.
2°. Oblate ellipsoids of gyration. In this case we have
The Y"-axes of the two hyperboloids now vanish. That of one sheet
becomes the part of the XZ- plane lying within the hyperbola
and that of two sheets the remaining parts. The points common to
both are those lying on the hyperbola, whose axes are "j/a2 — &2, ~|/ft2 — c2
and which passes through the remaining two foci of the system, and
is called the focal hyperbola. The axes of revolution of the ellipsoids
of gyration lie in the plane of the hyperbola.
75. Calculation of Moments of Inertia. In the case of a
continuous solid, the sums all become definite integrals, as stated
in § 68. All the preceding theorems of course are unaltered. If
the body is homogeneous all the integrals are proportional to the
density. Since the mass is likewise, the radii of gyration are in-
dependent of the density. We will therefore put Q = 1.
74, 75] CALCULATION OF MOMENTS OF INERTIA. 241
Rectangular Parallelepiped, of dimensions 2 a, 2b, 2c.
a b c
A'
—'a —b —c
a b c
'-///•
— a —b —c
a b c
"///•
— a — b — c
a b c
- c r r
C' = I I
€/ fj *J
— a — 6 — c
102) B = Cr + A' --= y abc (c2 + a2),
Cr = ^f + i?' = |-a&c
Thus the radii of gyration a07 60, c0 are
mo\
103) aQ=
Sphere, with radius E.
A' = CCCx2dxdydz, Bf =j f fodxdydz, C1 =
the limits of integration being given by the inequality #a-f y2+ 02c, B!=*-xb*ca, C' = ~xc*ab.
lo lo lo
105) B=C' + A' =
106) Q
Thin Circular Disk normal to Z-axis.
, C' = 0,
107) A = B = -
The moment about the normal to the disk is double that about a
diameter.
Circular Cylinder of radius JR, length 2L
The moment about the axis of rotation, is? as for the disk,
C = ~MR2, A = B' = ±
C'= fx
-i
108) A = B
75, 76]
We have
MOVING AXES.
if
243
Then the cylinder is dynamically equivalent to a sphere, as is
also the case for a cube.
These examples furnish the means of treating the cases that
usually appear in practice.
76. Analytical Treatment of Kinematics of a Rigid
System. Moving Axes. In §§ 55 — 57 we have treated the
general motion of a rigid system, from the purely geometrical point
of view, without analysis. We shall now give the analytical treatment
of the same subject. Let us refer the position of a point in the
system to two different sets of coordinates. Let %', y\ #' be its co-
ordinates with respect to a set of axes fixed in space, and let x, y, z
be its coordinates with respect to a set of axes moving in any
manner. The position of the moving axes is defined by the position
of their origin, whose coordinates referred to the fixed axes are
|, iq, g, and by the nine direction cosines of one set of axes with
respect to the other. Let these be given by the following table
X Y Z
The equations for- the transformation of coordinates are then,
109)
Since c^, a?, cc3 are the direction cosines of the X-axis with
respect to X', Y', Zf, we have
110)
and similarly
110)
a* + a,2 + ^ = 1,
244 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
Since the axes Y, Z are perpendicular, their direction cosines satisfy
the conditions,
ill)
and similarly,
i j .
Thus the nine cosines are not independent, but, satisfying six
conditions, may be expressed in terms of three parameters. These,
with the three |, ^, g, show the six degrees of freedom possessed
by a rigid system.
By interchanging the roles of the axes, and considering the
direction cosines of X', T, Z' with respect to X, Y, Z we find the
equivalent conditions
"I' + Ai' + yi'-l,
112) «22 + /322 + r22 = i,
113)
If we now differentiate the first of equations 109), supposing
y> % to be constant, we obtain
for the components in the directions of the fixed axes of the velocity
of a point fixed to the moving axes.
Let us now resolve the velocity in the direction which is at a
given instant that of one of the moving axes. To resolve in the
direction of the X-axis we have
115) Vx = tfX + C^Vy' + CC3V,' = ^ -^ + CC2 ~ + ttj ^
The coefficient of x in this expression is the derivative of the left-
hand member of the first of equations 110), and is accordingly equal
76] TRANSLATION AND ROTATION. 245
to zero. If we now denote the coefficients of y and z by single
letters, and compare them with the results of differentiating equa-
tions 111), writing
we obtain
** = a'§ + «• Tt + «» Ji
These equations express the fact that the velocity of a point attached
to the moving axes is the resultant of two vectors, one of which,
F, is the same for all points of the system, being independent of
x, y, 0, and having the components in the direction of xr, y', z' equal
*° dt' ~di' dt9 anc* *n ^e Direction °f %> y> z, equal to
TT dt- dr\ . d£
r*=tti-dt+a^t + K*dt'
118) ^-A + + '
This part of the motion is accordingly a translation.
The other part of the velocity, whose components in the direc-
tion of the instantaneous positions of the X, Y, Z-axes are given by
vx = qz- ry,
being the vector product of a vector o> whose components are p, q, r,
and of the position vector p of the point, is perpendicular to both
these vectors and is in magnitude equal to co Q sin (ra Q). It accord-
ingly represents a motion due to a rotation of the body with angular
velocity CD about an axis in the direction of the vector co. Thus we
have an analytical demonstration of the vector nature of angular
velocity. If we take as a position of the fixed axes one which
246 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
coincides with that of the moving axes at some particular instant
of time, the direction cosines vanish with the exception of a1} fl.2, ys,
which are equal to unity. We then have
-dt d dt --dt- dt
But since /33 = cos (?/£'), y2 = cos (#?/), we have on differentiating
d&* • f i\ d(iiz') dy« . f ,N d(zy')
-/f - - sm(^ ) T^fr' w=~?* ('y ) r&->
and since
sin(^) = sin(^)=l, f =
Thus it is clearly seen that p, q, r are angular velocities, being the
rates of increase of the angles #«/', x#', yx', or in other words, the
angular velocities with which the moving axes X, Y, Z are turning
about each other.
It is to be noticed that py q, r, though angular velocities, are
not time -derivatives of any functions of the coordinates, which might
be taken for three generalized Lagrangian coordinates q.
They are merely linear functions of the derivatives of the nine
cosines, which latter may themselves be expressed in terms of three g's.
If we seek to find those points of the body whose actual velocity
is a minimum, we must differentiate the quantity,
121) v* = (Vx+qz-ryy + (Vy + rx-p^ + (Vz + py-qxy
with respect to x9 y, z, and equate the derivatives to zero. We
thus obtain
r(Vy + rx— pz) — q(Vz+py — qx) = 0,
122) p(Vz +py - qx) -r(Vx + qz- ry} = 0,
q ( Vx + qz — r y) — p ( Vy + rx - pg) = 0,
which are equivalent to the two independent equations,
p q r
These are the equations of a line in the body, namely of the central
axis, as found in § 66, 38).
Calling the value of the common ratio A, clearing of fractions,
multiplying by p, q, r, and adding, we obtain the value of ^,,
n Y _L « T/ _i_ *.
124) i-*S
76, 77] RELATIVE MOTION. 247
Making use of this value of I with equations 121), 123), we obtain
for v for points on the central -axis
125) V =
2r
agreeing with 42).
If the velocity of points on the central axis is to be zero, we
must have
126) pvx + q.Vy + rV, = Q,
when the motion reduces to a rotation, as in 41).
77. Relative Motion. In forming equations 114) and the
following, we have supposed the point in question fixed in the body,
so that x, y, z were constants. If this is not the case we have to
add to the right hand members of 114) the quantities
dx . a dy . dz
^-st + ^^^dt'
IOTN dx . a dy , dz
127) *•-& + &-£ + **'
dx , „ dy , dz
"•di + hdi + v*di'
which, on being multiplied by the proper cosines, will appear in
equations 117) as -^j —_> -^> so that we have for the components
of the actual velocity in the direction of the axes X, Y, Z at the
instant in question, if the origin of the latter is fixed,
dx ,
v*Tjf + &"~ ry>
128) ^g+-«^
dz
These equations are of very great importance, for by means of them
we may express the velocity components in directions coinciding
with the instantaneous direction of the moving axes of the end of
any vector x, y, z. If for x, y, 2 we put the components of the
velocity v, we obtain the acceleration -components (§ 103), if the
components of angular momentum H we have a dynamical result
treated in § 84.
248 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION.
If we apply these equations to a point fixed in space, for which
vxf vy, vz vanish, we obtain
inc\\ dx dy dz
jj = ry-qz, -^=pz-rx, ~ = qx-py.
Taking a point on the X'-axis at unit distance from the origin, we
have x = cc1} y = ft , z = ylf
da* d&
and in like manner taking points on the Y' and Z'-axes,
da* dS* dy»
- - --
da, dp, dy.
— == ^r - ~> -j± from 130), we have
dp' dp Q dq dr
^-=«^ + Pi-di + Kdt>
iq9x dq' dp K dq dr
-dt-=a*-di + P*-dt+K^i'
dr' dp a dq dr
-df = ^dt+^M+^dt-
Thus the angular acceleration is obtained by resolving a vector whose
components about the axes X, Y. Z are ~ > -£> -.,-- In other words,
ctt dt cut
the time derivatives of the components p, q, r, of the angular velocity
in the directions of the moving axes at any instant are equal to the
angular accelerations of the motion about axes fixed in space
77, 78, 79] ENERGY AND MOMENTUM OF ROTATION. 249
coinciding in direction with the position of the moving axes at the
given instant. This theorem, which is by no means self-evident, is of
great importance, as is the similar property of the angular velocity,
of which we have already made use.
79. Kinetic Energy and Momentum due to Rotation.
From equations 119) we find for the part of the kinetic energy of
the rigid body due to the rotation, supposing -= - = -^ = ~ = 0,
dv dv ct v
- qrZmyz — rpZmzx — pqZmxy
and for the angular momentum, introducing 119) in 48),
Hx = Zm [y (py - qx) — z (rx —##)]= Ap — Fq — Er,
134) Hy = 2m\2(qz — ry} — x(py — qx)~] = — Fp -f Bq — Dr,
Hz = 2m [x(rx —pz] — y (qz — ry)] = — Ep — Dq + Cr,
the last column being what we obtained in § 68, 53).
It is evident that
o^s „ dT „ dT rr dT
Hx=^ ^V^Jq' Hz=frr'
so that in this respect p, q, r, Hx, Hy, Hz have the relation of
Lagrangian generalized velocities and momenta.
Since we have
the kinetic energy is one -half the geometric product of the angular
velocity and angular momentum.
250 ™. DYNAMICS OF ROTATING BODIES.
CHAPTER VII.
DYNAMICS OF ROTATING BODIES.
8O. Dynamics of Body moving about a Fixed Axis. The
simplest case of motion of a rigid body next to that of translation
is a movement of rotation with one degree of freedom, namely a
motion about a fixed axis. The centrifugal force exerted by the body
on the axis is Md&2 where d is the distance from the axis of the
center of mass of the body, and since this is in the direction of d7
which is continually changing, if a body is to run rapidly in bearings
the center of mass should be in the axis, otherwise the bearings are
subjected to periodically varying forces. At the same time, even if
this condition is fulfilled, there will be a centrifugal couple, also
tending to tear the body from its bearings, unless the axis is a
principal axis of inertia. It is worth noticing that the first condition
may be obtained in practice by statical means, by making the axis
horizontal, and attaching weights until the body is in equilibrium in
any position, but that the second condition is only obtained by
experiments on the body in motion. For this reason the former
condition is generally fulfilled in such pieces of machinery as the
armatures of dynamos, while the latter is not especially provided for.
Let us consider the motion of a heavy body about a horizontal
axis. The resultant of all the parallel forces acting on its various
particles is by § 59 a equal to a single force equal to the weight of
the body Mg applied at the center of mass. The position of the
body is determined by a single coordinate which we will take as the
angle # made with the vertical by the perpendicular from the center
of mass on the axis. If the length of the perpendicular is h the
work done in turning the body from the position of equilibrium is
The kinetic energy is
2) T =
a 6A
The equation of energy accordingly is
3) — K\-r:\ + Mgh(l — cos -fr) = const.
80] COMPOUND PENDULUM. 251
But this is the equation of motion of a simple pendulum of length
4) I- K --*2
~m "¥'
The body, which is often called a compound pendulum, accord-
ingly moves like a simple pendulum of length I. This is called the
equivalent simple pendulum. It is to be noticed that in virtue of
the constraint of rigidity, points at distances from the axis less
than I move more slowly than they would if moving alone in the
same paths, while those at greater distances move faster, and those
at distance I move just as they would if free to move in the same
circular paths.
Let kg be the radius of gyration of the body about a parallel
axis through the center of gravity. Then by § 70,
4) *» = ftJ + A«, Z = ? + ft,
ft
so that I is always greater than h. If we take a parallel axis 0' at
a distance h' = I — h beyond the center of mass 6r, so that it, 6r,
and the original axis are in the same plane, we have
5) hh' = 1$.
If now the axis 0' be made the axis of suspension, the equi-
valent simple pendulum has a length
fc'2 £2 _L h'*
6) V--hT = -~F~ = h + h' = L
The axis 0' is called the axis of oscillation, and we have the
theorem that the axes of suspension and oscillation are interchangeable
and separated by the distance equal to the length of the equivalent
simple pendulum. This is the principle of Eater's reversible pen-
dulum, used to determine the acceleration of gravity. The pendulum
is furnished with two knife edges, so that it may be swung with
either end down. Movable masses attached to the pendulum are so
adjusted that the time of vibration is the same in both positions,
and then the distance between the knife-edges gives the length I
from which g = -mr" The present example also includes the metro-
nome and the beam of the ordinary balance. The masses of the
pans may be regarded as concentrated at the knife-edges.
If the fixed axis is not horizontal, the modification in the result
is very simple. Suppose the axis makes an angle K with the vertical.
Let us take two sets of fixed axes, Z' vertical, Z the axis of rota-
tion, T1 horizontal in the plane of Z and Z', Y in the same plane,
252 VII. DYNAMICS OF ROTATING BODIES.
and X and X' coincident. Then we have for the transformation of
coordinates
z1 = — y sin a -f z cos a,
and determining the position of the system by the angle & made by
the perpendicular from the center of mass on the axis of rotation
with the Y-axis,
y = r cos #,
0' = — r sin a cos & -\- z cos a.
The potential energy is as before
W = Mgz' = — Mgli sin a cos # -f const.,
thus the equation of energy is
7) ' Y K \-TZ\ — Mgh sin a cos # = const.
Thus the equation is the same as before, except that the length of
the equivalent simple pendulum is increased in the ratio of 1 : sin a.
This example includes the case of a swinging gate and of the im-
portant physical instrument, the horizontal pendulum of Zollner.
The mode of action of the latter depends on the fact that the moment
of the force required to produce a given deflection #,
0 = --- = Mgh sin a sin #,
may be made as small as we please by decreasing a, which is
observed in practice by making the time of vibration long.
81. Motion of a Rigid Body about a Fixed Point.
Kinematics. We shall now consider one of the most important
and interesting cases of the motion of a rigid body, namely that of
a body one of whose points is fixed, and which thus possesses three
degrees of freedom. This case was dealt with very fully by Poinsot,
in his celebrated memoir "Theorie nouvelle de la rotation des corps",
in the Journal de Liouville, torn. XVI, 1851. On account of the
instructive nature of his processes, which are entirely geometrical,
we shall present his method first. The treatment of the properties
of the moment of inertia, which is contained in the same paper, has
already been given in §§ 70 — 72.
If one point of the body remains fixed, the instantaneous axis
must at all times pass through that point. The motion is completely
described if we know at all times the position of the instantaneous
axis in the body and in space, and the angular velocity about it.
80, 81]
BODY WITH FIXED POINT.
253
Fig. 71.
Let 0, Fig. 71, be the fixed point, and let OI± be the instan-
taneous axis at a given instant. During the time dt suppose a
line 0/2 moves to the position OJ2r, and
during the next interval /It let the body
turn about this line as instantaneous axis.
During this interval let another line OI3 move
to OJ3' which then becomes the instantaneous
axis, and so on. We have thus obtained two
pyramids, one Ol^OLOI^ . . . fixed in space,
the other O^OZ/OZ^. . . fixed in the body,
and we may evidently describe the motion
by saying that one pyramid rolls upon the
other. As we pass to the limit, making 4t
infinitely small, the pyramids evidently become
cones, and the generator of tangency is the
instantaneous axis at any instant.
The rolling cone may be external or internal to the fixed one.
In the former case, Fig. 72 a, the instantaneous axis moves around
the fixed cone in the same direction
in which the body rotates, and the
motion is said to be progressive, in
the second case, Fig. 72 c it goes in
the opposite direction, and the
motion is said to be regressive or
retograde. It is to be noticed that
it makes no difference whether the
rolling cone is convex (Fig. 72 a) or
concave (Fig. 72 b) toward the fixed
cone. (In the figures, in which
merely for convenience the cones
are shown circular, C± denotes the
fixed, C2 the rolling cone.)
If one of the cones closes up to a line, upon which the other
rolls, it always remains in contact with the same generator, that is,
the instantaneous axis does not move. Accordingly if either cone
degenerates to a line, the other does also, and the instantaneous axis
remains fixed in space and in the body. This case has been already
treated.
If we lay a plane perpendicular to the instantaneous axis at a
distance E from 0, Fig. 73, and if the radii of curvature of its inter-
sections with the fixed and rolling cones be Q± and Q2 (taken with
the same sign if they lie on the same side of the common tangent),
and the angles made by consecutive tangents at the ends of correspond-
ing arcs ds1 and ds2 are dri9 drif we have ds1
Fig. 72 a.
254
VII. DYNAMICS OF ROTATING BODIES.
The angle turned by the body in rolling the arc ds2 on its equal
ds1 is dr2 — drly and the angular velocity
> — — (d — fl ^ — — (— ~\
/1 1\
I
es Pj/
Fig. 72 c.
Now if w denote the angular velocity with which the instantaneous
axis is turning about an axis through 0 perpendicular to the common
tangent plane to the cones, we have
ds
which inserted in 8) gives
9)
Ql - Q.
If the cones have external contact,
$2 is negative, and if we consider the
absolute values, we must take the
sign plus.
Fig. 73.
or\
8') o? =
9') ^
Consequently if we give the values at every instant of three of
the quantities, co, the angular velocity about the instantaneous axis,
w, the angular velocity of change of the instantaneous axis, and ^
81]
FIXED AND ROLLING CONES.
255
Nfi
and p2, the radii of curvature of the sections of the fixed and rolling
cones, the fourth , and consequently the whole motion, are determined.
This corresponds to the fact that the body
has. three degrees of freedom.
If 01, Fig. 72, is the instantaneous
axis, OC± and OC2 the lines of centers of
curvature, the point I may be considered
to be travelling around the cones with the
ds
velocity -^-t and about the lines 00X and
002 as axes with the angular velocities v, ^,
of which 03 is the resultant. This was
proved by Poinsot as follows. Let Fig. 74
represent a section of Fig. 72 a in the plane
of the axes OC1; 01, OC2. If r19 r2 are
Pig. 74.
the perpendiculars from I on 001? 002,
and v and u are the angles C^OI, 02OJ, we have, considering absolute
values only,
10)
dt "*^~' T dt
1 dsl
cosu.
~^T = VCO$V, - -rr = II COS U .
g^ dt Q2 dt
Inserting in 8')
also since
we have
12)
09 = 4U- COS U + V COS V,
r, i\ ,
-~- = -r^—i and vr, =
*ITI 4t sm ^
It sin u = v sin v.
sm v sm w
That is, since the three axes are in the same plane, and G> is the
sum of the components of [i and v in its direction, while their
components in the perpendicular direction are equal and opposite,
a) is the resultant of ^ and v. Figs. 72 a, 72 b, 72 c show the three
cases, where the cones have external contact, where the fixed cone
is internal, and the rolling cone is internal, respectively. The
parallelogram construction is shown on the figures, and the direction
of rotation is shown by the arrows representing the vector rotations.
It will be noticed that in each case the arrow on the figure showing
the direction in which / is travelling around the rolling cone is
opposite to the direction of rotation /i about 02.
The rotation about OOj is known as precession. If both fixed
and rolling cones are cones of revolution, and /t, v, co constant, the
256 VII. DYNAMICS OF ROTATING BODIES.
precession is called regular. If we call # the angle C^OC* between
the axes of the cones, we have
, fl V CO
~^Tv ~ 53irT^ • im^
14) Co2 = jr + v2 + 2/iv cos ^.
An important case of a regular precession is furnished, us in the
motion of the earth, which, disregarding nutation (§ 93), describes
a cone with # = 23° 27' 32" in the time 25,868 years, the motion
being retrograde, Fig. 72 c. We thus have
sin 23° 27' 32" „„
smO ,
25.868x365.256
so that the pole of the earth describes a circular cone whose half
angle is 0",0087, an angle too small to be perceived by astronomical
means, the radius of the circle cut by this cone on the surface of
the earth being only 27 centimeters.
82. Dynamics. Motion under no Forces. We have already
found, §§ 68, 69; following Poinsot, the expressions for the momentum
and the centrifugal forces for the general motion of a rigid body.
If the fixed point be the center of mass, both the linear momentum
and the centrifugal resultant vanish, and we have to deal only with
the angular momentum and the centrifugal couple. At the same
time the resultant of the effect of gravity passes through the fixed
point, and is neutralized by the reaction of the support. Let us
then consider the motion of a body turning about its center of mass,
or more generally, the motion of a body under the action of no
forces. Such a motion will be called a Poinsot -motion.
Let OZ be the instantaneous axis. Then we have from § 68, 53)
Hx = - Em,
15) _H?, =
Let us call the resultant of Hx and Hy, J?2, Fig 75. We have for
the centrifugal couple Sc, from § 69, 59),
Lc = -
16) Mc =
Nc= 0.
Since Nc is zero, the axis of the centrifugal couple is perpendicular
to the instantaneous axis. But since
81, 82]
17)
CENTRIFUGAL COUPLE.
HxLe + HyMc + H2N0 = 0,
257
it is also perpendicular to the angular momentum. Consequently
the axis of the centrifugal couple
is perpendicular to the plane
containing the instantaneous axis
and the axis of angular momen-
tum, and is drawn in such a
direction that if 8C be turned
through a right angle in the
direction of the body's motion it
will coincide in direction with H2.
Also since the components of Sc
are equal in absolute value to the
components of H% multiplied
by co, we have
Fig. 75.
18)
Sc = HZ o = oH sin (Hci).
Thus the centrifugal couple is equal to the vector product of
the angular momentum by the angular velocity. In case the instan-
taneous axis is a principal axis at 0, the direction of H coincides
with that of « and the centrifugal couple vanishes. The body will
then remain permanently turning about the same axis. This property
of a heavy body turning about its center of gravity about a principal
axis of maintaining the direction of that axis fixed in space was
utilized by Foucault in his gyroscope, the axis of which points in a
fixed direction while the earth turns, and thus the motion of the
earth is made observable. The same principle is utilized practically
in the Obry steering gear contained in the Whitehead automobile
torpedo, in which a rapidly rotating gyroscope is made to give the
direction to the torpedo, and by acting on the steering gear to make
it return to its course if it accidentally leaves it.
Suppose on the other hand that the instantaneous axis is not a
principal axis. The centrifugal couple then tends to generate an
angular momentum whose axis is in its own direction, and this new
momentum compounds with that which the body already possesses.
Let us consider two successive positions of the body. Suppose that
in the time dt the body turns about the instantaneous axis through
an angle d(p = &dt. At the end of that time the vector H2 (Fig. 75),
would have turned through the angle dcp into the position H2, the
length of the infinitesimal vector H2H2 being H2dcp = H2&dt. But
during this time the centrifugal couple Sc has given rise to the
angular momentum
/
WEBSTER, Dynamics.
17
258 VII. DYNAMICS OF ROTATING BODIES.
This vector, being parallel to Se and thus perpendicular to H2'
gives, when compounded with J?2' a resultant exactly equal to H2.
The component of H parallel to co being unchanged by the motion,
we find, geometrically, that the angular momentum remains constant
throughout the motion, as we have found by a general theorem
in § 33.
As we now wish to follow the motion of the body from one
instant to another, it will be convenient to free ourselves from the
choice of axes which made the instantaneous axis the Z-axis. Let
us take for axes the principal axes of the body at 0. Let the com-
ponents of the angular velocity ro on the axes be p, q, r. Then the
angular momentum, being the resultant of the three angular momenta
due to the three angular velocities p, q, r, are by § 68, 53) or
§ 79, 134),
19) Hx = Ap, Hy = Bq, H, = Cr.
If we draw any radius vector to the ellipsoid of inertia at the
fixed center of mass
the perpendicular d on the tangent plane at the point #, y, g has
direction cosines proportional to Ax^ By, Cg.
If we draw the radius vector p in the direction of the instan-
taneous axis, so that
*>> ! = f = T = f = «>
equations 19) give
21) HM
or the angular momentum vector bears to the angular velocity vector
the relation, as to direction, of the perpendicular on the tangent
plane to the radius vector. Otherwise, if the angular momentum is
given, the instantaneous axis is the diameter conjugate to the
diametral plane of the ellipsoid perpendicular to the angular momentum.
The centrifugal couple being per-
pendicular to the plane of d and p,
lies in the diametral plane conjugate
to Q. It produces in the time dt an
angular momentum Scdt whose axis
is in the same direction. To find
the axis of the angular velocity
corresponding thereto we must find
the diameter conjugate to the plane
perpendicular to Sc, that is the plane yd. But the diameter conjugate
to a plane is conjugate to all diameters in it, hence the required
82] ROLLING POINSOT ELLIPSOID. 259
diameter is conjugate to Q and lies in the plane conjugate to Q, that
is, parallel to the tangent plane at x, y, a. Consequently, if we
compound with the velocity co about Q the velocity corresponding to
Scdt parallel to the tangent plane, the resultant has the same com-
ponent perpendicular to the tangent plane as co. In other words the
component to cos (eo, H) is constant throughout the motion.
Now we have found that H is constant in magnitude and direc-
tion, hence, multiplying by the constant ocos(ca#),
22) Ho cos (G)H) = const.
But jffcos (Ho) is that component of the angular momentum which
is parallel to the instantaneous axis, and is accordingly equal, by
§ 68, 53) to the product of the angular velocity by the moment of
inertia about the instantaneous axis.
23) H cos (Hm) = Ko.
Accordingly 22) becomes
24) K a)2 = const.
But this is equal to twice the kinetic energy. Accordingly we obtain
geometrically the integral of energy. Thus for a rigid body this
principle follows from that of the conservation of angular momentum.
In the ellipsoid of inertia we have, § 71,
Accordingly
25)
and the equation of energy shows that n is constant during the
motion, or during the whole motion the angular velocity is propor-
tional to the radius vector to the ellipsoid of inertia in the direction
of the instantaneous axis. But since cocos(i?(a) is constant, pcos(pd) = d
must be constant, and therefore the tangent plane is at a constant
distance from the center during the motion. But since the direction
of the line d is constant in space, and its length is also constant,
the tangent plane must be a fixed plane in space. As the point
where it is touched by the ellipsoid of inertia is on the instantaneous
axis the ellipsoid must be turning about this radius vector, and hence
rolling without sliding on the fixed tangent plane. The motion of
the * body is thus completely described, and we see that the problem
of a Poinsot- motion is equivalent to the geometrical one of the
rolling of an ellipsoid whose center is fixed on a fixed tangent
plane, together with the kinematical statement that the angular
velocity of rolling is proportional to the radius vector to the point
17*
260 VII. DYNAMICS OF ROTATING BODIES.
of tangency. Before taking up the discussion of this result, as given
by Poinsot, we will consider the analytical method of establishing
the result.
84. Euler's Dynamical Equations. If Hx', HJ, HJ represent
the angular momentum about the fixed X', Y', ^'-axes, L', M', N',
the moment of the applied couple, the equations of § 67, 49) are
dHf dH' dH'
<26\ _ £ _ TJ _JL — M1 — — N'
dt *"> dt *->. dt ~1V '
where (cf. § 76)
Hx=
27) HJ =
HJ = a,
Differentiating we have, after making use of § 77, 130),
dH' dH dH dH
28) -— + A + K
+ Hx (ft r-yiq) + S, fop -«,»•) + H, (at q - ftp).
If we now choose for fixed axes the instantaneous positions of the
moving axes, we have ax = /32 = y3 = 1, all other cosines zero, and
the equations 28) become simply
29)
dH
We may obtain the same results by the use of the equations § 77,
128). Let us take for the point x, y, z the end of the vector H.
Its coordinates with respect to the moving axes being Hx, Hy, Hz,
substituting them in equations § 77, 128) we obtain for their velocities
resolved along the X, Y, ^-axes the expression on the left of 29).
We must now put for Hx, Hy, Hz the expressions § 79, 134).
If now the moving axes are taken at random, the moments and
products of inertia of the body with respect to them will vary with
the time, so that their time -derivatives enter into the dynamical
equations, which are thus too complicated to be of any use. It is
therefore immediately suggested that we choose for the moving axes
a set of axes fixed in the body, and moving with it. The quantities
A9 B7 C, D, E, F are then constants. If in addition we take as axes
the principal axes at the origin of the moving axes, D, E, F vanish,
82, 84, 85] EULER'S EQUATIONS. 231
and then since Hx = Ap, Hy = Bq, H2 = Cr, the equations become
simply
30)
These are Euler's dynamical equations for the rotation of a rigid body.
In case the moments of the applied forces about the origin
vanish, they become
31)
and we see that the quantities on the right, being the vector product
of the angular velocity by the angular momentum, represent the
centrifugal couple, which alone acts to produce the angular accel-
eration, whose components appear on the left. We thus obtain the
result obtained geometrically by Poinsot, the quantities on the left
denoting the velocity of the end of H in the ~body.
The equations 29) may be simplified in another manner, if the
ellipsoid of inertia is of revolution. If for one of the moving axes
we take the axis of revolution, and for the others, any axes perpen-
dicular to it, whether fixed in the body or not, the axes will be
principal axes, and the moments of inertia constant, since the moment
of inertia about all axes perpendicular to the axis of rotation in the
same. Examples of this will be given in §§ 96, 106.
85. Poinsot's Discussion of the Motion. We may now
integrate the equations 31) by making use of the fact that the
centrifugal couple is perpendicular to the angular velocity and the
angular momentum. Multiplying equations 31) respectively by p, q, r
and adding
wHich is at once integrated as
33) 4 Ap* + 4 Bf + } Cr> = const.
This is the equation of energy.
262 VII. DYNAMICS OF ROTATING BODIES.
Multiplying now by Ap} Bq, Cr, and adding,
34) A*p% + B.q z = ~-
CO 00 0)
Now the length of the perpendicular d is, since it is the projection
of Q on the direction of the normal,
36) d = x cos (nx) -f y cos (ny) -f z cos (nz)
snce
2T -*««-£
Accordingly since T and H are constant, d is constant, and the
tangent plane being perpendicular to the invariable line H is fixed
in space. Poinsot called the locus of the pole of the instantaneous
axis on the ellipsoid, the polhode (rt6ho$ axis, bd6$ path), and its
locus on the tangent plane the herpolhode.
The ellipsoid of inertia being
37) Ax2 + W + Cz* = 1,
the distance of the tangent plane at x, y, z from the center is
Since this is to be constant, this equation with that of the ellipsoid
define the polhode curve. Combining the equation
39) A*x* + &f + C*s* = -~
with that of the ellipsoid, divided by d2, we obtain by subtraction
40) A -Ax* + B -Bf+ C - CS = 0.
85]
POLHODE AND HERPOLHODE.
263
This is the equation of the cone passing through the polhode, with
its vertex at the fixed point, that is the rolling or polhode cone.
We find then that the rolling cone for a body moving under
no forces is of the second order. If it is to be real, we must have
41) A^^C,
that is the perpendicular must have a length intermediate between
the greatest and least axes of the ellipsoid. If -^ = A the cone is
•42)
representing a pair of imaginary planes, intersecting in the real line
y = 3 = 0, the X-axis. Thus in this case the rolling cone reduces
to a line, fixed both in the body and in space. If ^ = C, we have
a similar result. If
we
43)
A(A- B)x* -
representing two real planes intersecting
in the Y-axis, and making an angle with
the XY- plane whose tangent is
44)
__
x ~ -
C(S-C)
These are the planes which separate the
polhodes surrounding the end of the major
axis from those about the minor axis. The
polhodes are twisted curves of the fourth
order, whose appearance is shown in
perspective in Fig. 77. The separating
polhodes are drawn black.
Since the polhode is a closed curve,
the radius vector of a point on it oscillates
between a maximum and a minimum
value. If 6 is the distance of a point
on the herpolhode from the foot of the
perpendicular d, since 62 = g2 — d2, 6 oscillates between two constant
values, and the herpolhode is tangent to two circles. Since the
polhode is described periodically the various arcs of the herpolhode
corresponding to repetitions of the polhode are all alike. The her-
polhode is not in general a reentrant curve. The name herpolhode
was given by Poinsot from the verb SQXSW, to creep (like a snake)
from the supposedly undulating nature of the curve, it has however
Fig. 77.
264
VII. DYNAMICS OF ROTATING BODIES.
been proved to have no points of inflexion, and is like Fig. 78, which
has been calculated for A = 8, B = 5, C = 3, ^ = 4 - 9.
86. Stability of Axes. We have seen that the body if
rotating about either of the principal axes of inertia will remain
rotating about it. If the instantaneous axis be the axis of either
greatest or least inertia, and be displaced a little, as the polhodes
encircle the ends of these axes the instantaneous axis will travel
around on a small polhode,
and the herpolhode will be
small, neither ever leaving the
original axis by a large amount.
These axes are accordingly
said to be axes of stable motion.
If on the other hand the mean
axis be the instantaneous axis,
and there is a slight displace-
ment, the axis immediately
begins to go farther and farther
from the original position, and
nearly reaches a point diame-
trically opposite before return-
ing to the original position.
pig 78< The mean axis is thus said to
be an axis of instability. It is
however to be noticed that if either A — B or B — C is small with
respect to the other, the separating polhode closes up about either
the axis of greatest or least inertia respectively, and thus a small
displacement may lead to a considerable departure from the original
pole, the rotation is thus less stable. The rotation about either axis
is most stable when the wedge of the separating polhode enclosing
it is most open.
87. Projections of the Polhode. From the equations of the
polhode 37), 39), we may obtain its projections on the coordinate
planes by eliminating either of the coordinates. Eliminating x,
45)
an ellipse of semi -axes,
B(A-B) S
the ratio of which is
yc(A-c\
85, 86, 87, 88]
PROJECTIONS OF POLHODE.
265
a constant, so that all the projections are similar. The motion about
the axis A is most stable when the small polhode is a circle, that
is when the above ratio is unity, or B = C.
Eliminating g we obtain
46) d2{A(A — C')x2 + B(B—C)y2}= 1 — <7tf2,
an ellipse the ratio of whose axes is
-i/B(B-C)
V A(A-C)'
and for maximum stability this is unity, or A = B. These projections
are shown in Fig. 79.
Fig. 79.
Fig. 80.
V,
Eliminating y, we have
47) 62{A(A — B)x* -
an hyperbola the ratio of whose axes is
C(B-C)
A(A-B)'
All the hyperbolas have the separating polhode projections as
asymptotes (Fig. 80).
88. Invariable Line. The invariable line describes a cone in
the body. Its equation may be simply found from consideration of
the reciprocal ellipsoid
4g) Z + F+Z?-1'
whose radius in the direction of d is -y and therefore constant. The
cone of the invariable axis is accordingly the cone passing through
the intersection of the ellipsoid 48) with the sphere
266
VII. DYNAMICS OF ROTATING BODIES.
49)
that is
50)
The axis of this, like that of the
polhode cone, is the axis of greatest
or least inertia.
Let us find how fast the invariable
line revolves around one of the principal
axes. Since the invariable axis is fixed
in space, its relative motion is equal
and opposite to the actual motion of
the part of the body in which it lies.
If we call A the diedral angle between
the plane of the invariable axis and
the axis of X and the XY- plane, we
may find -^- Projecting H upon the YZ- plane (Fig. 81), the pro
jection makes with the F-axis the angle A, given by
Fig. 81.
51)
from which
Differentiating,
52)
H
Cr
C / dr dq\ 1
BC
dr ^dq
Inserting from Euler's equations 31),
dq _ C — A dr _A-B
~dt ~ B r&' ~dt ~ C
dl p{B(A-B)q*+C(A-C)r*}
53)
dt
-H*
f\AH*-HP l]i = *\*H?-H
P
2AT
H*
-1
-EL\2
-©
sin2 (H x)
[88 MOTION OF INVARIABLE AXIS. 267
Similarly for the rotation around the T and Z-axes,
dv^ 3*0-1
dt ~ T sin2 (He)
Looking at the signs of the numerators, we see that the invariable
axis rotates around the axis of greatest moment of inertia in the
direction of rotation, about the least axis in the direction opposite
to that of rotation, and about the mean axis according to the
value of d.
If we mark off on the invariable axis a line of unit length, its
end describes a sphero-conic, the intersection of the invariable cone
50)
with the sphere
whose projections on planes perpendicular to the X and Z-axes are
ellipses, and perpendicular to the Y"-axis an hyperbola. The radius
vector of the X- projection is rx = sm(Hx) and since it turns with
the angular velocity ^- it describes area at the rate
55) 2%%£-*(^-i).
The time of one revolution of the body turning with the velocity p
would be, if p were constant, t = —
The equation of the ellipse is obtained by eliminating x from
the equations of the sphero-conic as
whose axes are
and whose area is
57)
'l
*
A
-,/*UB-JB
1
~A~
i
^
' F ^L-J5
fl
^2
A
1
A~
>2 — :
1
"0
0)
- K ^i-c
/ BC
' (A-B)(A-C)
268
VII. DYNAMICS OF ROTATING BODIES.
Now the area described in one revolution about the instantaneous
axis would be, if p were constant [see 55)],
and the number of turns the body makes for one revolution of the
invariable axis about the X-axis is the area 57) divided by this, or
59)
BC
-B)(A-C)
This may be made as large as we please by making A approach I>
or C. If B = C or the ellipsoid of inertia is of revolution, about
the X-axis, p is constant, and the invariable cone is circular, and
described with uniform velocity, the number of revolutions of the
body for one circuit of the invariable axis being /^_^\m The motion
is direct or inverse,
according as the
X-axis is that of
greatest or least
inertia.
These properties
may all be illustra-
ted experimentally
by means of Max-
well's Dynamical
Top1), constructed
by Maxwell for the
purpose of studying
the motion of the
earth about its
center of mass. An
example of this top
constructed in the
workshop of the
Department of Phy-
sics of Clark Uni-
versity is shown in
Fig. 82. The six
weights projecting
from the bell allow
Tig. 82. « .
the moments of in-
ertia to be changed
in a great variety of ways, while at the same time the center of
1) Maxwell, Papers, Vol. I, p. 248.
88] MAXWELL'S TOP. 269
mass is constantly kept at the point of support, a sharp steel point
turning in a sapphire cup. Maxwell's ingenious device for the
observation of the motion of the invariable axis, is the disk, divided
into four colored segments, attached to the axis of figure. The
colors chosen, red, blue, yellow and green, combine into a neutral
gray when the top is revolving rapidly about the axis of figure. If
however the top revolves about a line passing through a point in
the red sector, there will be in the center a circle of red, the
diameter of which is greater as the axis is farther from the center
of the disk and the boundaries of the red sector. Thus the center
of the gray disk changes from one color to another as the pole
moves about in the body, and by following the changes of color we
can study the motion. By noticing the order of the succession of
colors we can determine whether the axis of figure coincides most
nearly with the axis of greatest or least inertia, and by changing
the adjustments we may make it a principal axis, which is known
by the disappearance of wabbling, or we may make it deviate by
any desired amount from a principal axis. If the deviation is great,
and the top spun about the axis of figure, and then left to itself,
the top will wabble to a startling amount, but eventually the pole
will reach its first position and the wabbling will cease, to be repeated
periodically. The recovery of the top from its apparantly lawless
gyrations is very striking. If the adjustment is such as to make the
axis of figure lie near the mean axis of inertia, the top will
not recover, but must be stopped in its motion before striking its
support.
The path of the invariable axis has been made visible by
Mr. G. F. C. Searle, of the Cavendish Laboratory, Cambridge, by
attaching to the axis of figure a card, upon which ink was projected
from an electrified jet. Acting upon this suggestion, the author
attached to the top a disk of smoked paper, upon which a steel
stylus, playing easily in a vertical support (shown in Fig. 82 lying on the
table) could write with very slight friction. One easily finds by looking
at the disk in its gyrations a point which remains fixed, and by applying
the stylus to this point, holding it on a proper support, the path of the
invariable axis is drawn, and found to be an ellipse or hyperbola. If the
stylus is not held exactly on the invariable axis, small loops are formed,
which enable us to count the number of turns of the top in going
around the polhode, and thus to verify the theory. The results of
several spins are shown in Fig. 83, reproduced from actual traces.
The loops are turned out if the principal axis at the center of
the ellipse is that if greatest inertia, and in if it is the least, for the
reason that in the former case the invariable axis and the herpolhode
cone lie within the polhode (Fig. 83 a), while in the latter they lie
270
VII. DYNAMICS OF ROTATING BODIES.
without (Fig. 83 b) so that if we consider the relative motion, in the
former case a point fixed to the herpolhode describes a sort of
Fig. 88.
Fig. 83.
Fig. 88.
hypocycloid (loops out) on the card attached to the polhode, in the
latter a sort of epicycloid (loops in).
Fig. 83 a.
Fig. 83 a.
The recent astronomical discovery of the motion of the earth's
pole is probably due to a sort of variable Poinsot- motion, the moments
of inertia of the earth being gradually varied.
Cone
Fig. 83 b.
88, 89]
EXPERIMENTAL VERIFICATION.
271
Fig. 84.
89. Symmetrical Top. Constrained Motion. While we
have in the preceding section considered the very interesting and
instructive question of the motion of the most
general rigid body under the action of no forces,
by far the most frequent case under the practical
conditions of experiment is that in which the body
is dynamically symmetrical about an axis, that is,
the ellipsoid of inertia is of revolution. Such a
body we shall call a symmetrical top. This will
include not only all ordinary tops and gyroscopes,
as well as flywheels, rolling hoops, billiard balls,
but even the earth and planets. Suppose such a
body to be spinning under the action of no forces,
about its axis of symmetry. We have seen that it will remain so
spinning, and the angular momentum will have the direction of the
axis of symmetry. If now the axis
of symmetry OF (Fig. 84), is to
move to some other position, OF1,
which is then to coincide with the
new instantaneous axis, the angular
momentum HH' must be communi-
cated to the body, that is a couple
whose axis is parallel to HH' must
act on the body. This may be made
evident experimentally by placing a
loop of string over the axis F of
a symmetrical top balanced on its
center of mass (Fig. 85) and pulling
on the string. The axis of the top,
instead of following the direction of
the pull P moves off at right angles
thereto, although the string can only
impart a force in its own direction.
The pull of the string, together with
the reaction of the point of support
constitute a couple, whose moment
is perpendicular to the plane of the
string and of the point of support,
and it is in this direction that the
end of the axis, or apex of the top,
moves, as is required by the theory.
This simple experiment and the theory which it illustrates will make
clear most of the apparantly paradoxical phenomena of rotation. We
may describe it by saying that the kinetic reaction of a symmetrical
Fig. 85.
272
VII. DYNAMICS OF ROTATING BODIES.
rotating top is not at in the direction of the motion of the apex,
but nearly at right angles thereto. (Exactly at right angles to the
motion of OH.}
An ingenious application of this principle is found in the Howell
automobile torpedo, invented by Admiral Howell of the United States
navy. In this the energy necessary for driving the torpedo is stored
up in a heavy steel flywheel, weighing one hundred and thirty- five
pounds, and turning with a speed of ten thousand turns per minute.
The axis of the flywheel
lies horizontally perpen-
dicular to the axis of
the torpedo (Fig. 86), thus
steadying the torpedo in
its course. If now any
force acts tending to
deflect the torpedo hori-
se. zontally from its course,
by means of a moment
about a vertical axis, the end of the axis of the disk moves vertically,
causing the torpedo to roll instead of yielding to the deflecting force.
The rolling is utilized, by means of a vertically hanging pendulum,
to bring rudders into
action, and to cause
the torpedo to roll
back to its original
position, while main-
taining its course.
A striking example
of the principle
enunciated above is
found in an inge-
nious top (Fig. 87),
spinning on its center
of mass, with its axis
rolling on various
curves constructed of
metal wire. No matter
what the shape of the wire, the axis of the top clings to it as if
held by magnetism, no matter how sharply the curve may bend. The
passing around sharp corners at a high speed, in apparant defiance
of centrifugal force, is extremely remarkable. The explanation of the
action is immediate, on the lines just laid down. The instantaneous
axis passes though the point of support 0 (Fig. 88) and the point of
contact of the axis of the top with the wire. The wire, in fact,
Fig. 87.
89]
APPLICATIONS OF SYMMETRICAL TOP.
273
constitutes the directrix of the herpolhode cone. Since the ellipsoid
of inertia is of rotation, the axis of figure OF,
the instantaneous axis 01, and the axis of
angular momentum OH, lie in the same plane,
which is perpendicular to the tangent plane to
the herpolhode cone. During the rolling, all
these axes move parallel to this tangent plane,
so that the vector HHL, representing the change
of angular momentum, is parallel to the tangent
plane, and in the direction of advance of the
axis of figure. The couple causing the motion
accordingly due to the reaction between the
wire herpolhode and the top, is always parallel
to the tangent plane, and never vanishes, but
always tends to press the top against the wire.
Or in general, in constrained motion, the motion
causes the polhode cone to press against the herpolhode cone. This
seems to have been first explicitly stated by Klein and Sommerfeld,
Theorie des Kreisels, p. 173.
An application of the above
principle on a large scale, and the
only one known to the author, is
found in the Griffin grinding mill. A
massive steel disk or roller A (Fig. 89)
hangs from a vertical shaft by a uni-
versal or Hooke's joint C, in the middle
of a steel ring B forming the side of
a pan. If now the shaft be set rotat-
ing, the roller spins quietly about a
fixed axis, with no tendency to move
sidewise. If on the contrary it be
brought into contact with the ring,
it immediately rolls around with great
velocity, pressing with great force
against the steel ring or herpolhode,
and grinding any material placed in
the pan with great efficiency. It is
interesting to note that a somewhat
similar mill, in which the axis, instead
of passing through a fixed point, hangs
vertically from a revolving arm, and
therefore is devoid of the action just
described, although both mills possess in common the centrifugal
force due to the circular motion of the center of mass of the roller,
WEBSTER , Dynamics. 18
Fig. 89.
274 VII. DYNAMICS OF ROTATING BODIES.
is much less efficient. The first mill is an excellent example of the
centrifugal force and centrifugal couple, while the second lacks the
centrifugal couple, the instantaneous axis and the axis
of angular momentum being parallel.
Let us calculate the couple involved in the con-
strained motion involved in a regular precession, as
here applicable, in terms of the constants of § 81. If
the angular momentum make with the axis of figure
the angle a, its end is at the distance from the axis
of the fixed cone H sin (a + -91), so that it moves with
the velocity vH sin (a + #). This must be equal to the
applied couple,
60) K=vH$m(a + &).
Now resolving H parallel and perpendicular to the axis of figure,
we have
61) H cos a = Ceo cos u, H sin a = AM sinw,
so that
62) K= va(Asmucos& +
But we have, § 81,
-,o\ G> V
lo) — — - = —. — j
sin -91 sin u
14) a?2 = ^ -f v2 -
from which
ca sin u = v sin #, & cos u = [i — v cos #,
so that finally
63) K = v{Av8w&eo8& 4- (7 sin -9- (^-
It is to be noticed that the body will perform a regular precession
under no constraint or other applied couple, if putting .ZT=0,
n C
64)
C-A
9O. Heavy Symmetrical Top. We will now take up one of
the most interesting problems of the motion of a rigid body, namely
the motion of a body dynamically symmetrical about an axis, on
which its center of mass lies, and spinning about some other point
of that axis. This is the problem of the common top or gyroscope.
In order to determine the position of the top it will be convenient
to introduce three coordinate parameters, namely the three angles
of Euler. Let these be # the angle between the J^-axis, which
we take as the axis of symmetry, and the fixed vertical Zf-axis
89, 90]
HEAVY SYMMETRICAL TOP.
275
(Fig. 91). We may call the XT- plane the equator of the top.
Let ON (Fig. 91) be the line of nodes, or the line in which the
equator intersects the fixed
X' F'-plane. Let ij> be the
longitude of the line of
nodes, or the angle X'ON
measured positively from X'
to Y'. Let cp be the angle
from the line of nodes to
the X-axis, the positive
direction of increase being
from X to Y. By means
of the three angles -O1, ^, cp,
we may express the nine
direction cosines, and the
position of the body is
completely determined. The
meaning of the angles is
easily seen on the gyroscope
in gimbals (Fig. 92). It
will not be necessary for us to express the cosines, as we need
only the values of p, q, r in terms of Euler's angles and their velocities,
Fig. 91.
dt
dt dt
As these are the angular velocities about ON, OZ' and OZ, respectively,
we need only the cosines of the angles made by these lines with the
X, Y, ^-axes, which are evidently as given in the following table.
z]
N
Z
sin # sin tp
sin-O'COsqp
cos^
cosqp
— sin (p
0
0 •
0
1
Resolving now in the direction of the three axes, we obtain
65)
. - . .
= ~It Sm m ^ "^ ~dt COS *?'
dib .
18'
276
VII. DYNAMICS OF ROTATING BODIES.
These are Euler's kinematical equations. They illustrate the statement
made in § 76, about p, q, r as not being time derivatives, for it is
easily seen that pdt, qdt, rdt do not
satisfy the conditions of being exact
differentials.
The resultant of the weight of all
the parts of the body is Mg applied at
the center of mass. If this is at a
distance I from the fixed point the moment
of the applied force is Mglsmft about
the axis ON.
L= Mglsin&coscp,
66) M = — Mgl sin # sin cp,
N= 0,
so that Euler's dynamical equations are
A -~ = (B — C) qr -f Mgl sin # cos q, Hz = Cr on the vertical OZ', we obtain
70) HJ = Ap sin 0* sin (p -f Bq sin # cos cp -f- Cr cos # == const.
If the top is symmetrical about the 2T-axis, we have A = B.
Then the third equation 67) is
Cdr — 0
G~-°>
71)
Cr = const. = H2.
90, 91] EQUATIONS OF TOP. 277
The integral of energy 69) becomes
F + f* -jT = «-«cos#,
if we introduce the constants
7ox 2ft H* tMgl
-A- AC> a= -r-
The integral of vertical angular momentum 70) becomes
2
^4) sin # (p sin cp -f- q cos
TT dip
P» = H*'
88)
and the same force is required for the rotation about the vertical as
if there were no spinning, whereas a force is developed tending to
turn about the horizontal axis, which must be balanced by the
constraint, P$, proportional to -—• Thus the effect of the concealed
motion would be made evident, even if the disposition of the concealed
rotating parts were unknown. The effect of the gyroscopic term may
be described by saying that if the apex of the top be moved in any
direction, the spinning tends to move it at right angles to that
direction, as shown in § 50.
In our present problem, we have Py = 0,
^sin2# -^'4- C(
')cos# = const. = H},
or by 83),
89) A sin2 # • ^ + H, cos # = HJ,
which is the integral of 70).
The , differential equation for # is
90) A — Aty ' 2 sin # cos # + Hz sin # - ty ' = Mgl sin #,
which, on replacing ty' by the value from 89), and using the constants
of 73), 75) becomes
~^ - -_ a .
~*~ 8~ =S:
fj Qi
If we now multiply this by 2sin2# • ^r? it becomes an exact derivative,
and integrates into 77). Thus our three integrals are immediate
integrals of Lagrange's equations.
92. Nature of the Motion. Equation 78) which states that
the time -derivative of 2, the cosine of the inclination of the axis to
the vertical, is a polynomial of the third degree in 2, shows that 0
is an elliptic function of the time. As we do not here presuppose
a knowledge of the elliptic functions, we will discuss the motion
without explicitly finding the solution in terms of elliptic functions.
We see at once that the solution depends on the four arbitrary
constants cc, a, of the dimensions [T~2], which enter equation 78)
linearly, and /3, &, of dimensions [l7"1], which enter homogeneously
in the second degree, so that if we divide &, /3 by the same number
and a, a by its square, while we multiply t by the same number,
the two equations 78), 79) are unchanged, that is to any value of -0
280 ™. DYNAMICS OF ROTATING BODIES.
corresponds the same value of ^, or the path of the point of the top
is the same, but described at a different rate. Thus the shape of
the path depends on the three ratios of the constants, or there is a
triple infinity of paths. As for the meanings of the constants,
a depends simply on the nature of the top, irrespective of the motion,
and by comparison with § 80 is seen to be inversely proportional to
the square of the time in which the top would describe small oscilla-
tions as a pendulum, if supported with its apex downwards, without
spinning. If we change the top, we may obtain the same path by
suitably changing a, &, /3 as just described. These three constants
depend on the circumstances of the motion, b being proportional to
the angular momentum about the axis of figure, or to the velocity
of spinning, /3 to the angular momentum about the vertical, and
a depending on both the velocity of spinning and the energy constant.
Expressed in terms of the initial position and velocities they are
a=2^Z ^_C_
92) 0 =
With the convention that we have adopted, a is positive. As it is
evident that any path may be described in either direction, we shall
obtain all the paths if we spin the top always in the same direction.
We shall thus suppose b to be positive, while /3 may be positive or
negative, according to the sign and magnitude of (-5?) and cos #0.
dz \«*/o
Since -=- is real, f(si) 78) must be positive throughout the
motion, except when it vanishes. Since /"(I) = — (j3 — b)2 and
f(— 1) = — (|3 -f b)2 are both negative, f(oo) = oo and f(— oo) = — oo,
the course of the function
f(z) is as shown in Fig. 93.
Thus it is evident that f(z)
'Z has three real roots, two
#!, £2, lying between 1 and
- 1, while the third, #3,
lies outside of that interval
on the positive side. Thus the motion is confined to that part of
the #-axis between $if z2, and the apex of the top rises and falls
between the two values of <#• whose cosines are ^ and #2. The
triple infinity of paths may be characterized by giving the three
roots all possible real values, instead of giving the constants ft, /3, a.
In practice it will be convenient to give the two roots indicating
the highest and lowest points reached by the apex, and the value
92]
MOTION OF APEX.
281
of -~ the horizontal angular velocity at one qf them, which three
data completely characterize the motion.
Since g is an elliptic function of the time? the rise and fall is
periodic, and after a certain time, g will have attained the same
value, and so will -^ and -A accordingly during successive periods
the angles ^ and f(z} - A i (/?-K)2(i-*2)
™ TT?i**(A^:jrH 1_^2
We thus find that 0 — ^ is a factor of the expression on the right,
so that, multiplying by 1 — #2, we have f(z) exhibited in the form
where {^(z) is the polynomial of order two,
so that the other two roots are found by solving the quadratic
£(0) = 0. As the roots zl9 s^ approach each other, the rise and fall
decreases, and vanishes when f(z) has two equal roots. The condition
for this is that /(*) and f (0) = fa (0) + (0 — *i)fi(*) have a common
root a., that is that
from which
103) a(l-^)2
If % and one of the constants 6, /3 are given, this is a quadratic to
determine the other. We find
284 VII. DYNAMICS OF ROTATING BODIES.
which is constant, so that the motion is a regular precession, without
rise and fall. There are thus, for a given velocity of spinning, and
a given angle of inclination with the vertical, two values of the
velocity of precession. We may also find these by considering the
equation 90), putting & constant in which gives, if sin^ is not zero,
105) -4
a quadratic for ty1 with the roots
These values are real if 52 > 2acos#i. If the top be spun so fast
that -a?°S is a small quantity whose square may be neglected, we
find for one value of ty*
which is a large quantity of the order of 6, while the other root is
which is a small quantity of the order of y Of these it is the
slow precession which is usually observed.
It is to be observed that if we put ^' = v, (p* = /A, the first of
equations 82) gives for P$ the same result as obtained for K in 63).
When we make a vanish, so that the body is under the action of
no impressed moment, the root ^ becomes zero, so that the axis of
figure stands still, while the root ^ becomes - — that is, the body
performs a Poinsot -motion around the vertical as the invariable axis.
Thus the effect of the impressed forces may be looked upon as a
small perturbation of the Poinsot -motion.
We will now consider the motion when the condition 103) is
not fulfilled. From equation 78), we have t given by the elliptic
integral,
109) t=C- =^=
J ya(s-gj(g-sj(e-zj
We may easily find two limits within which this value lies, by
substituting for the factor ]/# — 03 in the integrand its greatest and
least values, as we did in the case of the spherical pendulum.
93] REGULAR PRECESSION. SMALL OSCILLATIONS. 285
Since throughout the motion
we have the inequalities,
110) C- dz > t > - - C-
By means of a linear substitution we may simplify the integral.
Let us put
/ 1 N Z1~ZZ
111)
when the integral becomes
112) C dz - = C dx = p.na— iff 4-
J y^-ax*-*,) J yi-x*~
so that we have for t,
113) — cos""1^ > t + const. > cos"1^.
If now the difference 8± — z% = x is sufficiently small in comparison
with £3 — 0J and #3 — ^2 , we may obtain an approximate result by
putting under the radical the mean of the quantities which are too
great and too small respectively, so that if ^ -f 22 = 2#0 we have
the approximate result
114) 1 4- const. = —cos-1x,
from which we obtain
x ^ c" " ~c = COS
115) ^ = ^0 + c • cos
The arbitrary constant has been taken so that i = 0 when the top
is at its highest, and z = £0 -f- c = £r
We thus see that when the roots zly ^ are nearly enough equal,
the apex of the top rises and falls with a harmonic oscillation g of
the small amplitude c = * ~ '• In order to determine when the
approximation is justified, we have to consider what will cause the
third root £3 to be large. Since #2 and #3 are the roots of the
quadratic function /i(#) 102), their sum is the negative of the coeffi-
cient of s divided by that of #2, that is
a h
286
VII. DYNAMICS OF ROTATING BODIES.
Thus we see that by making 1) large enough we may make #3 as
large as we please, when gl and #2 are given, so that the approxi-
mation is better the faster the top spins.
Let us now consider the horizontal motion, or precession. We have
117)
~dt
We have already supposed g to be a small quantity, so that if we
neglect the square of j— - — ^ we have, after developing the second
V1 ~ ZQ )
factor of the denominator,
118)
l-fefc,
dt
(1-V
Now inserting the value of g from 115) and integrating,
Thus we see that ifj varies with a harmonic oscillation about the value
that it would have in the regular precession at the mean height £0,
of the same period as the vertical oscillation. If we project the
motion of the apex on the tangent plane to the sphere on which it
moves, calling f; the horizontal coordinate, and tj the distance moved
from the horizontal mean axis, we have, Fig. 99,
120)
yi^v
Thus we see that the second terms of 115)
and 119) represent an elliptic harmonic motion
of the apex of the top. This is termed nutation.
We thus have a complete description of the
motion of a top when differing by a small
amount from a regular precession, as a regular
precession combined with a nutation in an
ellipse about the point which advances with
the regular precession.
We shall now make an additional supposi:
tion with regard to the constants of the motion.
We have seen from 108) that in the case of regular precession with
rapid spinning, the precession was slow. Let us then suppose that
121) •'=%— P
"•
is a small quantity of the same order as c, so that their squares and
product may be neglected. Since #0 is the cosine of the angle between
93]
NUTATION. CYCLOIDAL PATH.
287
the vectors whose magnitudes are b and ft this supposition is equi-
valent to saying that the angular momentum makes a small angle
with the axis of figure , as we see from
Fig. 100, in which the distance DE=p- bz0.
Making this supposition, the last term in
116) is negligible, also that in 119). Thus
we obtain from 116),
Jz «o
n
and since — is supposed to be large we may
neglect —
122)
123)
I, so that we have finally,
Fig. 100.
s*
— pi — T~ — ^smbt,
124) | = J^--°-*- ^si
Vi-^02 T/i-V
yr-
It is evident from Fig. 100 that /3 — is positive, accordingly
[cf. 119)] the apex is always moving so that -~ is positive at the
bottom of its path, and thus the average motion is in that sense.
The motion at the top may be in either direction, according to the
magnitude of c. We see that the motion of nutation is opposite to
the motion of the clock -hands. Thus the motion of the apex, as
given by 124), is that of a point at a distance — from the
• yi-V
center of a circle which rolls on a line above it with its center
advancing at a velocity - — - — • The radius of the rolling circle is
- ~V
Such a locus is called a cycloid. In the ordinary cycloid, the
tracing point is on the circumference of the rolling circle, or
/3 — = be. If the tracing point is an internal one, the cycloid
is called prolate. It has no loops, nor vertical tangents, and -^ is
never zero, but it has points of inflexion. If the point is external
the cycloid is called curtate, and has loops, but no inflexions. It is
evident that this curve will be described when the apex is given a
push to the left at the top of its motion, while if it be given a
push to the right it describes the prolate cycloid, and if it be simply
let go, it describes the ordinary cycloid with cusps. (The prolate
and curtate cycloids are also called trochoids.) Since the height of
a cycloid is to the length of its base as 1 : x, the base being the
288
VII. DYNAMICS OF ROTATING BODIES.
distance traversed in one revolution, we see that when the top is
spun rapidly, so that the precession is slow, the rise and fall is very
/ c*\
rapid (for b = r -j-j ? and very small. For this reason it is seldom
noticed, and this accounts for the popular opinion, expressed in many
text hooks, that the motion of a top is such that its axis describes
a circular cone with a constant angular velocity, or a regular
precession. Thus the reason of the vertical force of gravity producing
a horizontal motion remains a paradox. We have seen that such a
motion is the very particular exception, and not the rule, being only
exhibited when the necessary horizontal velocity is imparted at the
outset, so that the action of gravity is always balanced by the
centrifugal couple generated by the precession. If the necessary
velocity is not imparted, the top immediately begins to fall in
Fig. 101 a.
Fig. 101 b.
obedience to gravity. The motion which we have just described is
called by Klein and Sommerfeld a pseudo- regular precession, and may
be called a small oscillation about
a regular precession. In Fig. 101
are shown curves of the actual
path obtained by photographing
a small incandescent lamp attached
rig. 101 c. to the axis of a gyroscope, with
0- nearly a right angle.
94. Small Oscillations about the Vertical. In the discussion
which has just been given, it has been supposed that 1 — #2 was
not a small quantity. If however in the course of the motion the
axis of the top becomes nearly vertical this will no longer be true,
so that for this case a special investigation is necessary. Let
us suppose that & and #' are so small that in the kinetic potential
all their powers above the second may be neglected. Let us use for
coordinates the rectangular coordinates of the projection of the apex
on the horizontal plane,
x = r cos ^', y = r sin ^, r = sin #.
Using then the expression of 85) for the kinetic potential, with
W= Mglcosft,
93, 94] TOP NEARLY VERTICAL. 289
125) ^ = ~ A(&'2+ sin2 » - $' 2) + H, cos «• • ^ - Mgl cos #,
we will convert it into terms of #, ?/, #', ?/', neglecting all terms of
order higher than the second.
In the first term, since , to the order of approximation,
r' = cos# •»' = #',
we have rt2 -f- ^2^'2, the square of the velocity in polar coordinates,
which is in rectangular coordinates x'2 + y'*. Also we have
11 9 , ,• xy' — yx'
tan ib = ~> sec2^-^f = — — j2 — >
iC iC2
126) ^' = ^#'
and since
127) cos a = { 1 - (*2 -f 2/2)}¥ = 1 - ^jp^
we have finally
128) $=i^'H
We have then in the term in H, an example of the gyroscopic
terms of § 50, in which x = q±, y = g2,
Forming the equations of motion, since
dy dx
we have finally
Ax" + H2y' + Mglx = 0,
Ay"-H,x' + Mgly = 0,
or in terms of our constants,
131)
These equations are a particular case of a problem that is interesting
enough to he considered in full. If & were zero, they would be the
WEBSTER, Dynamics.
290 V11- DYNAMICS OF ROTATING BODIES.
equations for the small vibrations of a system of two degrees of
freedom , the stiffness and inertia coefficients of which are the same
for both freedoms. Let us consider the general system , for which
132) T = ±(Ax"+By'*), W= ±-(Cx* + Df),
into which a gyrostat, or rapidly rotating symmetrical solid , is intro-
duced ; the direction of whose axis is determined as in the present
case by the coordinates x and y. (It is to be noticed that x and y
are principal coordinates.) The equations for the small oscillations
of the system are then
By" - Hzx' + Dy = 0.
These may be treated by the general method of § 45 for small
oscillations. In order to simplify the notation, it will be convenient
to put
134)
, A, ,
when our equations become
8" + &,' + <*-<>,
V'-fcl' + cfy^O.
Having solved these, we may pass to the case of our vertical top
by putting c = d.
In accordance with the method of § 45, let us put
from which we obtain
^(*2 + C)
-4&A
The determinantal equation is
137) A4 + (c
whose roots are
138) A2 = {-(c +
If the solution is to represent oscillations, all the values of A must
be pure imaginary, thus both values of ^2 must be real and negative.
SMALL OSCILLATIONS ABOUT VERTICAL. 291
If we call them — ^2, — v2, we have for their sum and product the
coefficients in 137),
139) ii* + v* = c + d + tf, [i*v* = cd.
In order that p, v shall be real it is accordingly necessary that c, d
shall be of the same sign, that is our system must be either stable
for both freedoms , or unstable for both. Extracting the square root
of the second equation 139) , doubling, and adding to and subtracting
from the first,
* v2 = c d b* 2~/cd = ¥ ~
Extracting the roots, adding and subtracting,
141)
The inner double sign is evidently unnecessary. Since fi v = +
we have also,
From the values of [i and v it is evident that both are real if c
and d are positive. If tehy are negative it is necessary in order to
have real values that
b > ~^c + y^d.
Thus we find that even if the system is unstable, sufficiently rapid
spinning of the gyrostat makes it stable. This is the case of the
top with its center of mass over the point of support. In order to
complete the discussion we have to determine the coefficients Alt A^
for the various roots. If we call the roots
19
292 VII. DYNAMICS OF ROTATING BODIES.
we have for the general solution ,
| = 41' e"" + Af e~if" + Af e"" + A(? e'1",
n = 41' «"" + Af e-"" + Af e1" + Af e~ivt,
where we have by the first of equations 136) ,
41' .„_„• 2) ' 3) « >
1 /LA 1 " -— - •> _ ~
>* 1~ '
Introducing the values of the J.2's in terms of the A,? a, and writing
145) 41)+42)=«, 41)-42)=-ift43)+44)=«', 43)-A^=-in',
we have, replacing exponentials by trigonometric terms,
| = « cos lit + (Ism [it + «' cos v^ 4- /3f sini/tf,
146) e _ u.2 c —
?? = -- (j3 cos ^t — asi
with the four arbitrary constants a, ft «', /3', or putting
147) a = ^ cos sl9 /3 = ^ sin £1; a' = A% cos «2; /3' = A2 sin
| = Al cos (^^ — fj) -f -^2 cos (v^ ~ ^2)7
148) ft2-
sin t - f -- sin v -
Accordingly the motion may be described as the resultant of two
elliptic harmonic motions of frequencies ^? ^7 the directions of the
axes of the ellipses being coincident, and given by the directions in
which the system can make a principal oscillation when the gyrostat
is not spinning. The absolute sizes of the two ellipses are arbitrary,
but the ratios of the axes, and the phases, are determined by the
nature of the system and the rapidity of the spinning.
Calculating the coefficients in 148) from the values of /it, — > v, — >
149)
bv
94]
EXPERIMENTAL REALIZATION.
293
If now c = d, as in the case of the top, both these expressions
become equal to plus or minus unity, so that both ellipses become
circles. The motion of the top
making small oscillations about
the vertical is accordingly to be
described by saying that its apex
describes epicycloids (epitrochoids)
or hypocycloids (hypotrochoids)
upon the horizontal plane. It is
to be noticed that according as
we take the signs in 149) the
relative sense of the rotations in
the two circular motions will be
alike or different. By considering
which way the top tends to fall
we may decide whether the cusps
are turned inwards or outwards,
and it will be found that if the
center of mass is above the point
of support the cusps or loops
are turned inwards, and the curves
are epicycloids, while if it is
below the cusps or loops are
turned outwards and the curves Mg 102
are hypocycloids.
An instrument to show these properties of the motion has been
constructed by the author, and is shown in Fig. 102. A heavy
Fig. 103 a.
Fig. 103 b.
symmetrical disc hangs by a universal joint (Hooke's or Cardan's
suspension) from a shaft which is rotated by an electric motor.
294
VII. DYNAMICS OF ROTATING BODIES.
A pointed steel wire slides easily in the end of the axis of the pen-
dulum, and draws a curve upon a plate of smoked glass which is
Fig. 103 c. Fig. 103d.
brought against it by a lifting table: By means of a lantern and a
Fig. 103 e. Fig. 103 f.
right angled prism the curves are projected upon a wall in the act
Fig. 103 g!
Fig. 103 h.
of being traced. Examples of the curves obtained are shown in
Fig. 103. (Figs, g, h, i are hypocycloids drawn geometrically, for
94]
COMPARISON OF THEORY AND EXPERIMENT.
295
comparison.) In order to compare theory with experiment, let us
calculate how many revolutions in one circle
go to one of the other. Let us call this
ratio m. We have then from 141)
150)
m = ±
V
Fig. 1031.
It is noticeable that this ratio depends only
on the constants of the system and the
velocity of spinning, hut not on the circum-
stances of projection. This is shown in the
figures. In each group m is made an integer,
by properly adjusting the height of the disc, and the rate of spinning,
which is main-
tained constant by
stroboscopic ob-
servation. If the
apex is merely
drawn aside, and
let go, the curves
have cusps. If
pushed to one
side, the curves
have loops, and
if to the other,
there are no loops,
but the curve is
a sort of curvi-
linear polygon,
and if the spinning
is rapid enough,
there are in-
flexions. The ratio
m is the same for
the three types of
curve. The slight
perturbations no-
ticeable in the
figures arise from
the slight loose-
ness in the tracing
point, and permit p.g 1Q4
of counting the
number of revolution of the top about its axis (thus determining r),
296
VII. DYNAMICS OF ROTATING BODIES.
which is found to be the same for the same value of m, as may be
verified on the figures.
In order to illustrate the more general case above treated, the
spinning top is included in a system of two pendulums (Fig. 104),
whose frequencies may be made to have any ratio to each other, so
that when the top is not spinning the point describes a Lissajous's
curve. The influence of the spinning on the curves is shown in Fig. 105.
Fig. 105 a.
Fig. 105 b.
An interesting application of the heavy symmetrical top is the
gyroscopic horizon invented by Admiral Fleuriais of the French navy.
A small top is spun upon a pivot in vacuo, in a box which is
attached to a sextant. The top executes a slow movement of precession
Fig. 105 c.
Fig. 105 d.
about the vertical, and by means of lines ruled on two lenses which
it carries, the vertical is observed, so that observations may be made
when the horizon is obscured by fog.1)
1) Schwerer. L'horizon gyroscopique dans le vide de M. le Contre-Amiral
Flewiais. Annales Hydrographiques. 1896.
" ! T
94, 95] APPLICATION OF JACOBFS METHOD. 297
^^^^•HMfiflW*^^^
95. Top Equations deduced by Jacotai's Method. We will
conclude the treatment of the top by deducing the equations of
motion by the method of Jacobi, § 41. Since we have for the kinetic
energy,
77) T=
and for the momenta
p# = Aft', py = J.sin2# • ^' 4- C cos 0- (9'
p(p = C(, ty, plus a function & of -9-, which we will determine.
We shall obtain the result in the notation of § 90 if we, put
154) S=-ht + A(bg> + ^ + ®).
nserting in the differential equation 153), we obtain
155)
from which
156)
Accordingly we have the solution,
157) S = --ht + A (by + ^
The integrals are obtained by differentiating by the arbitrary constants,
*, 6, ft,
298 VII. DYNAMICS OF ROTATING BODIES.
dS
Bearing in mind that -Fffr) = . vL> and that -- = r, we see that
Sill $T O
the first equation is the integral of equation 78), the second of 79),
and the third of 80).
96. Rotation of the Earth. Precession and Nutation.
Since the earth is not an exact sphere, it is not centrobaric, that is
the direction of the resultant of the attraction of its various parts
on a distant point does not pass through its center of mass. Or, in
other words, the attraction of a distant mass -point, not passing
through the center of mass of the earth, possesses a moment about
it, which tends to tilt the earth's axis. The sun and moon are so
nearly spherical that they may
^.^ be considered as concentrated
• •< — r~/\ a^ their respective centers of
4<^_J mass. One of them, placed
Fig. 106. at M (Fig. 106), attracting the
nearer portions of the earth
more strongly than the more distant ones, tends to tip the earth's
axis more nearly vertical in the figure, and it is seen that this is
the same in whichever side of the earth the body lies. Thus the
sun always tends to make the earth's axis more nearly perpendicular
to the ecliptic, exept when the sun lies on the earth's equator, that
is at the equinoxes. The deflecting moment thus always tends to
cause a motion of precession in the same direction, the tendency
being greatest at the solstices, and disappearing at the equinoxes.
The moon, which moves nearly in the plane of the ecliptic, produces
a similar effect.
It will be shown, in § 148, that the potential of a body at a
distant point, x, y, 8 is given very approximately by
J, - --
- -2 - -pr-
where r2 = x2 -f y2 + £2, and A, B, C, are the principal moments of
inertia of the body. If the distant point is the center of the sun,
whose mass is m, the force exerted by the earth on the sun is
\zr\\ v 2F T/ dV dV
X = rm-, Y=ym, Z
95, 96] ROTATION OF THE EARTH. 299
But this is equal and opposite to the action of the sun upon the
earth, the moment of which about the earth's center of mass is
accordingly
161) M=-(eX-xZ),
Differentiating the expression 159), since x appears both explicitly,
and implicitly in r, and -^- = — ?
J dx r
dv^dv^
dx dr r
162) |F = |I^ +
oy or r
fa — +
cV dV z_
r
and inserting in 161),
L = -
163)
We may now insert these in Euler's equations, so that, if x, y, z,
the coordinates of the sun, are given as functions of the time, the
earth's motion may be found. Considering the earth to be symme-
trical about its axis of figure, we put A = _B, so that N = ,0, and
the third equation gives r = const., as in the case of the top. It is
however more convenient for our purpose to use, instead of Euler's
equations a set of equations proposed by Puiseux, Resal, and Slesser,
in which we take for axes, as suggested in § 84, the axis of symmetry,
and two axes perpendicular to it, that is, lying in the equator, and
moving in the earth. We have, since we are dealing with principal axes
164) Hx = Ap, H, = Aq, Hz=Cr,
which are to be inserted in equations 29), § 84, where we are to'
put the velocities with which the moving axes turn about themselves,
which we will call ^0, g0, r0, so that our equations are
165)
dH,
dt
dH
dH
300
VII. DYNAMICS OF ROTATING BODIES.
If we choose as X-axis the line of nodes, or intersection of the
equator with the ecliptic, or plane of the sun's orbit about the earth,
we have, in 65), cp = 0, so that Euler's geometric equations become
simply,
1 a while r is n°t equal to rQ. Inserting
in the third equation 165), we have C— = 0, r = const. = &, where
i& is the angular velocity of the earth's daily rotation.
We shall content ourselves with an approximate solution of the
equations, which may be obtained by neglecting the squares and
products of small quantities. Observations show that -^ and -r- are
/ dib \
small, l-j-j- = 50",37 per year), so that we may neglect r0g0,
Thus our equations 165) become
167)
If the sun, or other disturbing body, did not move with respect to
the axes of X, Y", Z, then Z, M would be constant, and the equations
would be satisfied by constant values of j?, #,
168)
M
In order to ascertain whether these approximations are sufficient
when L and M vary, let us differentiate equations 167), substituting
in either the value of the first derivative of p or q from the other,
obtaining
169)
dt*
A
CO,
A
^.T^
4- M) =
dL
~dt"'
dM
dt
Fig. 107.
We have now to find the values of L, M
in terms of the motion of the sun.
If I be the longitude of the sun, that
is the angle its radius vector OS makes
with the X-axis, we have, passing a plane
through the sun perpendicular to the X- axis
(Pig. 107),
x = r cos I, y = r sin I cos #,
£ = r sin I sin -fr,
96] PRECESSION AND NUTATION. 301
so that, inserting in 163),
jj = 7 — L_ — 1 sm2 1 sm ft, cos ^
170)
Ti/r 3ym(A— C} . ^ 7 .
M = ~ ' sin 6 cos Z sin &.
If we suppose the sun's path relative to the earth to be a circle,
described with angular velocity n, we have
so that
171)
d L &yin (C A) \
3ym'(A—C) f 07 . . 7 ^d&
rs" I*'1 COS ^ Sm ^ + sm * cos ^ COS # -TT
dt
Now if A = C, there would be no motion of the earth's axis, so
that C — A is a small quantity of the order of -=J • The angular
velocity n, though much larger, is still 365^- times smaller than i&,
so that if we neglect its product and that of ^- with C — A, we
may neglect the right hand sides of 169). Thus the approximation 168)
is justified, for differentiating, it will make ~ > -jrf negligible, so
that equations 167) are satisfied. Inserting the values of p, q, L, M,
in 168), we have
(C~ ^cosfrd -cos2r>,
These are the equations for the precession and nutation. In order
to integrate them approximately, we may neglect the small difference,
on the right, between # and its mean value, so that inserting the
value of 21 = 2nt -\- 210, considering & constant, and integrating,
3ym C - A _,
* - ifir* "(T COB*' "
J 3ym C-J. ,,
* = sm *
We thus find the motion to be a regular precession, of amount
t rr ,j\ i 3ym (7— ^4.
174) * =db^r-cos*'
together with a nutation in an ellipse (compare § 93), whose period
is one -half that of the revolution of the disturbing body.
302 VH. DYNAMICS OF ROTATING BODIES.
By means of observations of the value of the precession, we
tQ ^\
may thus obtain the ratio of - — ~n~^' We see that ^ne forces causing
precession are proportional to —s- On account of the nearness of the
moon, therefore, and in spite of its small mass, the precession
produced by the moon is somewhat greater than that due to the
sun. Since the moon's orbit departs but little from the plane of the
ecliptic the precession due to the moon may be calculated approxi-
mately by the above formulae, and compounded with that due to
the sun.
97. Top on smooth Table. Having treated in detail the
motion of a body with one point fixed, and three degrees of freedom,
it remains to consider the motion of bodies which, like the ordinary
top, spin upon a table or other surface. We must now consider
the reaction between the body and the surface, and we have to
distinguish between the ideal case of perfectly smooth, or frictionless
bodies, where the reaction is normal, and bodies between which there
is friction, so that the reaction is not normal. We will consider the
first case. Let us examine the motion of a symmetrical top, spinning
on a sharp point resting on a smooth horizontal plane. The top has
five degrees of freedom, its position being defined as before by the
three angles -O1, ^, (p, and in addition, by the coordinates x, y, of the
center of mass, the #- coordinate being given by
0 = I COS ft.
Since the only force which we have not already considered is
the reaction, which has no horizontal component, the horizontal
component of the acceleration of the center of mass vanishes, so that
its motion is in a straight line with constant velocity. It therefore
remains only to determine the motion of rotation. This being in-
dependent of the horizontal motion just found, we may consider the
latter to vanish, so that the center of mass will be supposed to move
in a vertical line. The motion thus becomes one of three freedoms,
and we shall treat it by Lagrange's method as before. By the
principle of § 32, 50), the kinetic energy is equal to that which the
body would have if concentrated at its center of mass,
plus that which it would have if it performed its motion of rotation
about the center of mass supposed at rest. If then A and C denote
the moments of inertia about the center of mass (in § 90 they were
the moments about the fixed point), we have
175) T=~ [M Z
96, 97, 98] TOP ON" TABLE. FRICTION. 303
The potential energy is as before Mglcosfr. Consequently the only
difference in the problem from that treated in § 90 is in the extra
term in #', Ml* sin* & • &'2 in the kinetic energy. Carrying out the
various steps of §§ 90, 91, we find instead of the first equation 76)
the equation
176) #
and putting s = cos #,
d
(d*\*—
\dtf-
where we denote the roots of the denominator by #4, z5. It is to be
noticed that they lie outside the interval 1, — 1, for evidently the
coefficient of #'2 in 176) cannot vanish for real values of -9-.
The square of -j-- being now the quotient of two polynomials
in z, s is a hyperettiptic function of t. We may however, without a
knowledge of these functions, treat the problem just as we did the
former one, and we shall find that the top in general rises and falls
between two of the roots of the numerator, and that the motion
resembles the motion already discussed. The path of the peg has
loops, cusps, or inflexions, according to the initial conditions, as
before, while the regular precession and the small oscillations may
be investigated as before. Whereas accordingly the functional relations
involved are considerably different, physically this motion, which is
that of the common top, closely resembles that already studied.
98. Effect of Friction. Rising of Top. We have now to
take account of the effect of friction. Here we have in addition to
the normal component of the reaction a tangential component called
the force of friction, and the ordinary law assumed is that the
tangential component is equal to the normal component multiplied
by a constant depending on the nature of the two surfaces in contact,
called the coefficient of friction. If the friction is less than a certain
amount, the two surfaces will slide one upon the other, and the
direction of the friction will be such as to oppose the sliding, being
in the direction of the relative motion of the points instantaneously
in contact. The bodies are then said to be "imperfectly rough". If
the friction is greater than a definite amount, it will prevent the
sliding, and there is then no relative motion of the points of contact,
so that there is a constraint due to the friction, which is expressed
by an equation stating that the velocities of the points of the two
surfaces in contact are equal. If one of the surfaces is at rest, as
is usually the case, the instantaneous axis then always passes through
304 VII. DYNAMICS OF ROTATING BODIES.
the point of contact. If it is in the tangent plane, the motion is
said to be pure rolling, and the bodies act as if "perfectly rough".
If the instantaneous angular velocity has a normal component, this
is known as pivoting, and is also resisted by a frictional moment.
The pivoting friction is however usually neglected where the surfaces
are supposed to touch at a single point. The conception of perfect
roughness, involving the absolute prevention of slipping under all
circumstances is as far from the truth as that of perfect smoothness,
nevertheless slipping may often cease in actual motions, so that
motions of perfect rolling, whether or not accompanied by pivoting,
are important in practice. For instance, a bicycle wheel under
normal circumstances rolls and pivots, if it slips the consequences
may be serious.
In the following sections we shall consider the methods of treating
various cases of friction. We may however, without calculation,
consider the effect of imperfect friction on the motion
of the top spinning on the table. Let P (Fig. 108)
represent the peg, no longer considered as a sharp
point. Let OH represent the angular momentum at
the center of mass 0. The friction is in the direc-
tion Fj opposite to the motion of the point of
contact of the peg with the table. The moment of
this force with respect to the center of mass is
perpendicular to the plane 01$ or K. Thus the end
of OH moves in the direction of K, that is rises.
Thus the effect of friction is to make the top rise
toward a vertical position. When it has reached
that, it "sleeps" and the friction has become merely
pivoting friction, tending to stop the motion. We
have before seen that under conservative forces, the top would never
become vertical except instantaneously by oscillation.
The effect of friction on the Maxwell top may be most easily
seen from the fact that the friction tends to stop the spinning,
accordingly it causes a moment which is represented by a vector
opposite in direction to ro, Figs. 83 a, b. Compounding this vector
with H we see that the moment of momentum vector H tends to
move away from the axis of the two cones in Fig. 83 b while it tends
towards it in Fig. 83 a, thus the trace of the invarible axis (as it
would be but for friction), instead of being an ellipse, is a spiral
winding outwards in the former case, and inwards in the latter, as
is shown by the arrows in Fig. 83.
99. Motion of a Billiard Ball. We wiU now treat the
problem of the motion of a sphere on a horizontal plane, taking
98, 99] MOTION OF BILLIARD BALL. 305
account of friction. The friction of sliding is supposed to be a force
of magnitude F= pR where R is the reaction between the ball and
the table, and ^ the coefficient of friction. F has the direction
opposite to that of the motion of the point of contact of the ball
and table.
If the axes of X, Y are taken horizontal , Z vertical, we have
for the motion of the center of mass the equations
~ = X = Fcos (Fx) = [iR cos (Fx),
178)
and since 0 is constant, R = Mg.
Euler's dynamical equations are, since A = B=C,
179)
a being the radius of the sphere. To determine the direction of JE
we have
IT V
180) m f-^=,
where vx, vy are the velocities of the lowest point of the sphere,
Differentiating these equations, and making use of 179),
*vx _d*x d^_^J_ai^
'dt~W a dt~ M" A2
182> S-3+.Jf-i+5'- •
Dividing one of these by the other, and using 180),
dvx dvy
20
from which
WEBSTER, Dynamics.
306 VH. DYNAMICS OF ROTATING BODIES.
Integrating we have
184) ^ = const. = ^-
Thus we find that F makes a constant angle with the axes of
coordinates , and since it has the constant magnitude pMg the center
of the sphere experiences a constant acceleration, and describes a
parabola.
If the center of the sphere starts to move with the velocities
Vx, Vy and with a "twist", whose components are ^>0, g0, r0, we have,
integrating 179), since X, Y are constant,
185) q = qo-a^t,
•
r = r0.
Integrating the equations for the center of mass
Inserting in 181) we find for vx, vy
i Xt>
~V 2
187) M
V -£
^"=r = =T
Accordingly,
X Vx-aq0
188) ^^-g, -/X2+r2 = ^,
F^-««o
X = - ^M^r
189)
Since vX9 vy are linearly decreasing functions of the time, whose
ratio is constant, they vanish at the same time
t
99, 100] PURE ROLLING. 307
The sliding the ceases. Obviously it cannot change sign, so that the
above solution ceases to hold. The ball now rolls without sliding,
and we have always, at subsequent instants, the equations of constraint
dx dy
Y = &y _ n*p_ _ <* V
M dt* dt " A
From this we obtain
so that X = Y = 0, and the ball moves uniformly in a straight line.
In reality there is always a certain friction of pivoting, causing
a moment about the normal, but this would only affect the rotation
component r, which would not affect the motion of the center of
the ball.
1OO. Pure Rolling. The preceding problem has illustrated
both sliding friction and pure rolling. The treatment of the latter
is interesting on account of a peculiarity in the nature of rolling
constraint which makes the ordinary treatment of Lagrange's equations
require modification. We shall accordingly first present the application
of Euler's equations to this subject, but before doing so, we will
treat by means of results already obtained one of the most important
practical problems, which illustrates the steering of the bicycle, namely
the rolling of a hoop or of a coin upon a rough horizontal plane.
As the hoop rolls, if its plane is not vertical, it tends to fall,
and thus to change the direction of its axis of symmetry. The falling
motion developes a gyroscopic action, which causes the hoop to pivot
about the point of contact, so that the path described on the table
is not straight but curved. The pivoting motion, like the precession
of the top, tends to prevent the falling, and to this is added the
effect of the centrifugal force due to the curvilinear motion of the
center of mass. Thus the hoop automatically steers itself so as to
prevent falling, and a bicycle left to itself does the same thing.
Let the position of the hoop be defined by the coordinates of
its center of mass, and by the angles #, ^,9 of § 90, # being the
inclination to the vertical of the axis of symmetry, or normal to the
plane of the hoop at its center. We will examine the conditions for
a regular precession, in which <$•, qp', ^' being constant, the center of
mass and the point of contact of the hoop with the table evidently
20*
308
VII. DYNAMICS OF ROTATING BODIES.
describe circles. In this case we have for the moment about the
center of mass of the forces tending to increase &, by 82),
191) P^ =
The forces which act to change # are, the weight of the hoop,
which has zero moment about the center of mass, and the reaction
of the table. Let i?, Fig. 109, (an edge view of
the hoop) represent the vertical reaction, F the
horizontal component due to friction, which is
normal to the path of the point of contact, the
tangential component disappearing on account of
the assumed constancy of the velocity of rolling,
as in the case of the rolling sphere. We accord-
ingly have, taking moments,
192) P<> = Fa sin # - Ea cos #,
F a being the radius of the hoop. But considering
rig. 109. the motion of the center of mass, which is uniform
circular motion, and supposing all the forces there
applied, since there is no vertical motion, the resultant vertical
component vanishes, or E = Mg and the horizontal component balances
the centrifugal force, so that
193) F =
where b is the radius of the circle described by the center of mass.
Beside the dynamical equation we have the equation of constraint
describing the rolling. Since there is no slipping, the rate at which
the center of mass advances in its path is
194) ar = a(y1 + ^' cos#).
But this is also, from the circular motion, equal to —biff'. From
the equation of constraint,
195) a(
200) A f - (Cr -
dr
We have finally, as the conditions for rolling and pivoting, the
equations stating that the velocity of the point of contact with the
plane (whose coordinates are x,y, 8) is at rest.
vx 4- qz = 0,
201) vy + rx - pz = 0,
vz — qx = 0.
The coordinates x, s of the point of contact are obtained as known
functions of & from the equation of the meridian of the hody. We
have accordingly the eighteen equations 197 — 201) between the
eighteen quantities
vx, vy, vz, p, q, r, pQ, g0, r0, X, Y, Z, #, ^, qp, Ex, Ey, E2,
or just enough to determine them. The differential equations are all
of the first order.
The reactions may be at once eliminated from the equations
199), 200). By differentiating 201) we may eliminate the derivatives
of vX9 vy, vz from 199). In doing this, however, we introduce the
derivatives of x, 2, which are functions of #, so that in general the
equations become complicated. We shall therefore confine ourselves
to the case of a body rolling on a sharp edge, like a circular
cylinder with a plane bottom, or a hoop or disk. We then have
x, z constants,
x = a, z = c,
where a is the radius of the circular edge, c the distance of the
center of mass from its plane, which is zero in the case of the hoop.
100] GENERAL MOTION OF ROLLING HOOP. 311
The equations of the motion of the center of mass thus become
M ( — c TJT ~l~ aq2 ~\- PQ (Q>T — CJP)) = RX ~}~ Mg sin -91,
OAON n/r I dp dr \ -n
zOZ) Mi c~ — a~j cro giving
Aajl + Cc~ + Car + Aapcin& = 0,
207)
- Ma*p + Macpctnfr = 0,
312 VII. DYNAMICS OF ROTATING BODIES.
as two equations to determine p and r as functions of #. When
they are thus determined, the equation of energy
208) M(v2 + v2 + v^+A(p2+q2) + Cr2 = 2{h-Mg(a8m&-ccos&)},
or by 201),
209) (A + Mc2}p2 + (A + Ma2 + Mc2}q2 + (C+ Ma2)r2 - ZMacpr
suffices to determine q as a function of #. Thus we see that when-
ever & returns to a former value, the circumstances of the rolling
are repeated, so that the motion is periodic.
Eliminating -~ from 207), we obtain
210) MAa2p = - {AC + M (Aa2 + Cc2}} |£ - MCacr,
differentiating which, we may eliminate p, obtaining for r,
0.,-,\ d*r , _ dr . MCa
211)
a linear differential equation for r, with variable coefficients. In the
case of the disk, where c = 0, by introducing the new variable
x = cos2 #,
we reduce the equation to the form
,. x d*r . /I 3 \ dr
212)
which is the differential equation of Gauss
213) ^(l-^l^ + ^-
if we put
MCa
This differential equation is satisfied by the hypergeometric series
214) F(K,ft,r,x^i + +
tt(« + i)(« + a)p(/? + i)(/? + 8) 3
and by the theory of linear differential equations we find that the
general integral is
-f
100, 101]
LAGRANGE'S EQUATIONS AND ROLLING.
313
where ct and c2 are arbitrary constants. From this we obtain
Ma* dr
* ~ Ma*~ d&'
•j f\ ' -t t
and from equation 209) we obtain a = -=-> or -=- as a function of #,
at d&
so that the time is given in terms of # by a quadrature. The
explicit completion of the solution is too complicated to be of use
in investigating the motion. The equations 207) have been investigated
by Carvallo by a development in series, from which the properties
of the motion are investigated.
1O1. Lagrange's Equations applied to Rolling. Noii-
iutegrable Constraints. In the attempt to apply the method of
Lagrange to the problem of rolling we are met with a peculiar
difficulty, which has been the subject of researches by Vierkant and
Hadamard.1) We shall follow the treatment of the former of the
rolling of a disk. Let us characterize the position of the disk by
the angles ^, ^, - & -f cos^sin^ • ^'),
220) /n' = y' - a (sin # sin ^ • -9-' — cos ^ cos ^ • ^'),
£' = a cos # • #'.
Squaring and adding, we obtain for the kinetic energy,
221) T= \
+ 2a[— si
-h cos # • ^' (— x' sin ijf -f- y' cos ?/>)]}
-f |^(#'2+ sin2#- ^'2) -f |C(y + T^'cos^)2.
J I
Forming now the equation of d'Alembert, adding equations 216)
multiplied by A and ^ respectively, and equating to zero the coeffi-
cients of dx, dy, d&9 Sty, 4^, we obtain the equations of motion
101] NON-INTEGRABLE CONSTRAINTS. 315
o •
e) C(q)' 4- ^'cosd-) -
We must observe that if we had taken account of the equations 217)
in the expression 221) for the kinetic energy, before differentiating,
we should have obtained quite different equations. Having performed
the differentiations, however, and introduced in the equations all the
reactions belonging to the different coordinates, we may now take
account of the equations of constraint, thus introducing, in effect,
the statement of the equilibration of some of the reactions, and
causing some of the terms to drop out.
Now introducing the values of #', y\ from 217) in 222), and
eliminating A, \L from 222 a, b, e), we obtain
223) Ma2 1 sin ^ -j- (sin ^ • ') 4- ^sin2# •
4- Jf a2 sin «••«•>' = 0.
316 VII. DYNAMICS OF ROTATING BODIES.
Since the last terms of both these equations contains &', it is suggested
that we change the independent variable from t to #, which is done
by dividing through by #', giving
226a) (Ma2 -f C) ~ (yf -f cos # - ^) - Ma2 sin & • tf = 0,
b) ^{(Ma2+ C) cos &
dz
vz = yp-xq + -ft-
The first two terms, representing the vector -product of the angular
velocity of the moving axes hy the position -vector of the point,
represent the components of the velocity of a point fixed to the
moving axes, the last terms represent the velocity relative to the
moving axes.
We might now, in order to find the components of the actual
acceleration along the instantaneous positions of the moving axes,
make use of equations 128), § 77, to obtain the velocity of the end
of the velocity- vector, that is put for x,y,z the quantities vx,vy,vz,
when on the left we should ohtain ax, ay, az, as has been suggested
for H in § 84 (after 29) but we shall rather choose for the sake of
variety, to proceed by means of Lagrange's method to find the forces
tending to increase the relative coordinates x, y, z. Suppose a particle
of mass m to have coordinates x, y, z in the moving system. Its
kinetic energy is then
that is
r- !«[($'+ ®'+
-A -=^> the angular velocities
Gnf d t (it
of the moving axes p, q,r and their derivatives, ^ty ~> ~
u>\ ut Ci/t
A point fixed to the moving system at x, y, 8 would have the
accelerations •
d
r * >
236) 0*0 = a - * - y (r2 +^>) + q (rz+px),
These may he called the components of acceleration of transportation
(entramemenf) or the acceleration of the moving space. They represent
the centripetal acceleration of the transported point. (If p, q, r are
constant, we have in the last two terms the ordinary expressions for
centripetal acceleration, whose resultant is v* divided by the distance
from the axis of rotation.) Beside these and the relative accelerations
there are terms
T <>\ndz vdy
-di -rdi
These are termed the components of the compound centripetal accel-
eration. We accordingly have for the total acceleration
'x,
238)
dt
that is the actual acceleration of the point is the resultant of the
relative acceleration, the acceleration of transportation, and of the
compound centripetal acceleration. Accordingly we may consider the
axes at rest if we add to the actual forces applied forces capable of
producing an acceleration equal and opposite to the acceleration of
transportation and the compound centripetal acceleration. This is
known as Coriolis's theorem.
320 VII. DYNAMOS OF ROTATING BODIES.
The resultant J, often known as the acceleration of Coriolis, is
evidently perpendicular to the relative velocity whose components
are -jrr> ~> -^ and to the axis of jp, q, r and is equal to twice the
vector-product of the angular velocity of the axes and the relative
velocity of the particle. It is interesting to notice that the accel-
eration of Coriolis arises from the presence of linear terms in the
velocities, -^y ^| > -^ in the kinetic energy, the effect of which in
introducing gyroscopic terms was explained in § 50. Thus a particle
may be arranged to represent hy its motions relatively to a uniformly
revolving body, such as the earth, the motions of a system containing
a gyrostat. This remark is due to Thomson and Tait.
1O4. Motion relatively to the Earth. Let us suppose the
axes chosen are taken fixed in the earth, the origin at the center,
the #-axis the axis of rotation. Let the earth rotate with the constant
angular velocity <£, which expressed in seconds is
and is very small. Then p = q = 0, r = &. The centripetal accel-
eration of transportation is then
Accordingly for a point at rest on the earth we may consider the
earth at rest, provided we add to other applied forces a centrifugal
force whose components are m&2x, m&2y. This centrifugal force is
239) ro^yS^Tp = m&2Rco8 y,
where E is the radius of the earth and q) is the latitude. This is a
subtractive part of g, the acceleration of gravity, which is consequently
greatest at the poles, least at the equator. The vertical part of the
centrifugal force is m&2Rcos2cp. This acceleration is common to
all bodies at rest on the earth, and hence is included with gravity
in our ordinary experiments. It need not then be further noticed.
There is however to be considered the apparent compound centrifugal
force, — m Jx, — mJy, — mJz, which acts on bodies in motion rela-
tively to the earth.
- mJx=
24°) ;
- m J, = 0.
103, 104] EFFECT OF EARTH'S ROTATION. 321
The equations of motion of a body acted on by forces X} F, Z are
m
dt
241)
the terms in & having the usual property of gyroscopic forces. For
a falling body we have, if the plane XZ is the vertical plane at the
place of observation,
X = — mg cos qp, Z = — mg sin -i/8 (£0 - £)3
^ ~ ~3 V ~ g
104, 105] FOTJCAULT'S PENDULUM EXPERIMENT. 323
The particle falls to the east by an amount proportional to the square
root of the cube of the height of fall and to the cosine of the
latitude. This has been experimentally verified.
1O5. Motion of a Spherical Pendulum. We have for the
pendulum the equation of constraint
so that to the previous equations of motion are added terms
giving
245)
Multiplying by -^i ~t -^ respectively and adding, then integrating,
we get the equation of energy,
the gyroscopic terms disappearing, as usual. For a second integral
we get as in § 23,
If we assume that the oscillations are infinitely small,
t i-
is infinitely small, and the last term above is of the third order and
may be neglected. Integrating we have
248)
The equation of energy 246) becomes
Inserting polar coordinates,
| = r cos o,
= r sn o,
21
-j. = b — SI sin
r =
while if to be the velocity of rotation of the top about the ^-axis,
we have for the moment of momentum, the axes being principal
axes, though not fixed in the top,
255)
Hx = — A& sin #, Hy = A
Hz = Co.
~ dt
Inserting now in the equations 29), § 84, the constraint producing
a couple Z,
d&
~dt
256)
+
n do
Q -77 —
at
.
--J-- -f
dt
~
-^- = 0.
dt
From the last of these equations, o is constant, while from the
second, neglecting i£2, we have
The first equation 256) determines the constraint L. Equation 257)
is the equation for the motion of a plane pendulum, § 22, so that
the gyroscope will perform oscillations about a line parallel to the
earth's axis, or will be in equi-
librium when -91 = 0, thus afibrd-
ing a means of determining the
latitude. The time of a small
oscillation will be, 2
which, on account of the small-
ness of &, will be very great
unless o be made very great.
The experiment was performed
with success by Foucault.
In the second case let us
suppose the gyroscope con-
strained to move in a horizontal
plane. Let us take for IT- axis
the vertical, corresponding to
the £-axis of Fig. 112, for the Z-axis the axis of figure of the
top, making the variable angle (p with the north, towards the
east, and for the X-axis a perpendicular to these (Fig. 113). The
rotation of the earth gives the components — & sin #, & cos # in the
direction of the £, g axes respectively (# being the co- latitude and
Fig. 113.
326 Vn. DYNAMICS OF ROTATING BODIES.
not variable), which give by the use of the table of direction cosines,
X
r
z
i
— sinqp
0
— cosqp
n
— cosqp
0
sing?
t
0
i
0
the values of the rotations of the axes
p = SI sin # sin op,
QCD\ f) CL drp
T = £i sin # cos op,
and for the angular momenta, o being again the velocity of spinning,
Hx = A& sin # sin op,
o r" i"\\ ~TT A i 4~\ d ($)\
Hz = Co.
Inserting in equations 29), § 84, the constraint producing the couple Z,
4 f~\ * tt CD y^f / f~* (t QP\
yl 5i sin v cos op -T— -j- C o ( 5i cos v" — ~^^ I
- J. te cos 0- - ~\ £1 sin ^ cos op = L,
d2cp
260) — A -=TJ + ^15i2 sin2 ^ sin op cos op — C&& sin ^ sin op = 0,
C-~ -\- A ( 5i cos ^ - ~p ] 5i sin -O1 sin op
— A& (£1 cos & — -=?] sin ^ sin op = 0.
V «£/
The last equation again shows that o is constant, while from the
second, neglecting £i2 we have
261) -v-^ -j~ -j" »&£> sin -&• • sin op = 0.
The first equation determines the constraint L. The gyroscope again
performs oscillations about the meridian, with the period
A
which is greater the greater the latitude, being infinite at the poles. The
gyroscope in this mounting therefore constitutes a dynamical compass.
It is to be noticed in both cases that the equilibrium is stable
for # = 0 or qp = 0 if o is positive, and for & = it, cp = it if o is
negative, in other words the gyroscope tends to set its axis as nearly
as possible parallel with the earth's axis, so that its direction of
rotation shall correspond with that of the earth. This was clearly
stated by Foucault, although he employed no mathematics.
PART III
THEORY OF THE POTENTIAL,
DYNAMICS OF DEFORMABLE
BODIES
107, 108] POINT -FUNCTION. LEVEL SURFACE. 329
CHAPTER VIII.
NEWTONIAN POTENTIAL FUNCTION.
107. Point -Function. If for every position of a point in a
region of space t a quantity has one or more definite values assigned,
it is said to be a function of the point, or point -function. This
term was introduced by Lame. If at every point it has a single value,
it is a uniform function. Functions of the two or three rectangular
coordinates of the point are point -functions. A point -function is
continuous at a point A if we can find corresponding to any posi-
tive £, however small, a value d such that when _B is any point
inside a sphere of radius < d,
\f(B)-f(A)\ q3 be three uniform point -functions. Each has a level
surface passing through the point M. If these three level surfaces
do not coincide or intersect in a common curve, they determine the
point M, and we may regard the point -functions qlf q2, q^ as the
coordinates of the point M. The level surfaces of qly q2, qB are the
coordinate surfaces, and the intersections of pairs (q1q2)) G&fe), (Q'sQ'iX
are the coordinate lines. The tangents to the coordinate lines at M
are called the coordinate axes at M. If at every point M the co-
ordinate axes are mutually perpendicular, the system is said to be
an orthogonal system.
110. Differential Parameter. The consideration of point-
functions leads to the introduction of a particular sort of derivative.
If F is a uniform point -function, continuous at a point M, and
possessing there the value F, and at a point M' the value F', in virtue
of continuity, when the distance MM' is infinitesimal, F' -V=JV
is also. The ratio y, _y ^y
is finite, and as MM' — As approaches 0, the direction of MMf
being given, the limit
108, 109, 110]
DIFFERENTIAL PARAMETER.
331
lim
AV _dV
As ~ Js
is defined as the derivative of V in the direction s. We may lay off
x} T7"
on a line through M in the direction of s a length M Q = -*— and
OS
as we give s successively all possible directions,
we may find the surface that is the locus of Q.
Let M N (Fig. 115) be the direction of
the normal to the level surface at M, and
let MP7 drawn toward the side on which V
is greater, represent in magnitude the derivative
in that direction. Let M ' and N be the inter-
sections of the same neighboring level sur-
face, for which V=V, with MQ and HP.
Then
AV
AV MN
MM' MN MM'
As MM' approaches zero, we have
lim
AV
dF
MM'
,. AV dV
lim -v-,= 5—;
en
MN
Hence
oV
ds
that is, the derivative in any direction at any point is equal to the
projection on that direction of the derivative in the direction of the
normal to the level surface at that point. Accordingly all points Q
lie on a sphere whose diameter is M P.
The derivative in the direction of the normal to the level surface
was called by Lame l) the first differential parameter of the function V,
and since it has not only magnitude but direction, we shall call it
the vector differential parameter, or where no ambiguity will result,
simply the parameter, denoted by P or Py. The above theorem
may then be stated by saying that the derivative in any direction is
the projection of the vector parameter on that direction. The theorem
shows that the parameter gives the direction of the fastest increase
of the function V.
If V is a function of a point -function #, F= /*(#), its level
surfaces are those of g, and
and if
T~> dV- dVdq /•»/ \$<7
P == __ = *. = f' (Q) _i,
^^ dq on ' *•*? on
dn
k, P =
1) G. Lame. Lemons sur Us coordonnees curvilignes et leurs diverses appli-
cations. Paris, 1859, p. 6.
332 VIII. NEWTONIAN POTENTIAL FUNCTION.
where the sign + is to be taken if V and q increase in the same,
- if in opposite directions.
Suppose now that V = f(ql} q%, q3, • • •)
d V d V 0 q^ d V d q% d V 3 qs
and if h±, h2, . . . denote the parameters of qly #2, . . . the above
theorem gives
T> /T^ N 2V 1 /7 \ , 2F7 n \
P cos (Ps) = o — h* cos (his) + o — ^9 cos (h»s) H
0 ql v q% •
dV
Now + -o— hi is the parameter of V, considered as a function
of fa, and we may call it the partial parameter Pl} and since P/
and hi have the same sign if -g— > 0, opposite signs if g—- < 0, we
have in either case
-o— hi cos (his) = Pi cos (PiS).
0%
Pcos (Ps) = P! cos (P^) + P2 cos (P2s) H
This formula holds for am/ direction s and therefore shows that
the parameter P is the geometrical sum, or resultant, of the partial
parameters,
T> ~p I -p I
Thus we have the rule for finding the parameter of any function
of several point -functions. If we know the parameters hlth%, . . . of
the functions ql9 q2, . . . and the partial derivatives g — > g — > • • • we
lay off the partial parameters
Q Y*-
in the directions hlf h2, . . . or their opposites, according as ^— > 0,
or the opposite, and find the resultant of Plf P2, . . .
If the functions qlt q2, ... are three in number, and form an
orthogonal system, the equation
gives for the modulus, or numerical value of the parameter
Examples. (1) in § 108. Let the distance of M in the given
direction from the plane be u. 4V=4u = ~- > where a is the
cos a
angle between the given direction and the given plane.
p_
An cos a cos a
110] - DIFFERENTIAL PARAMETER INVARIANT. 333
If the given direction is perpendicular to the given plane P = 1.
Accordingly for q{ = x, q2 = y, qB = 2, the rectangular coordinates of
a point, we have Px = Py = Pz = I, and for any function f(x, y, e)
,dx) - \dy) " {-$*,
The projections of P on the coordinate axes are the partial parameters
This agrees with the definition already given in § 31.
Consequently, if cos (sx), cos (sy), cos (ss) are the direction cosines
of a direction s, the derivative in that direction
~— = P! cos (sx) -f -Pg cos (sy) + -^3 cos (S;r)
0F , v , 01? , v . aF / x
= - cos s« + cos $ -f cos s^,
which is the same as equation 38 a) of § 31.
We have in this section defined the differential parameter in a
geometrical manner, not depending on the choice of axes of coordinates.
If however we take as the definition of the arithmetical value of the
parameter the equation
P =
and then transform to other coordinates x1, y',0', by equations 109),
§ 76, we easily find by calculation that
w,
is equal to P, that is, the parameter is a differential invariant, as is
at once evident from its geometrical nature.
If f(x, y, z) is a homogeneous function of degree n, by Euler's
Theorem,
or
nf= P{xcos(Px) + ycos(Py) + ^cos(P^)}.
Now the ± parenthesis is the distance from the origin of the
tangent plane to the level surface at x,y,z. Calling this d,
nf=±P*, P = ±*f>
334
VIII. NEWTONIAN POTENTIAL FUNCTION.
or the parameter of a homogeneous function is inversely proportional
to the perpendicular from the origin to the tangent plane to the
level surface. For example, if n = 1,
7 = ax -r by -f cz,
The level surfaces are parallel planes, and the parameter is
constant,
V is proportional to the distance of the level surface from the origin.
If n = 2,
as
P=2y£+£+ii
For the surface, F= 1,
a familiar result of analytic geometry.
111. Polar Coordinates. If we call the point functions of
Examples 2, 3, and 4, of § 108, r, #, ) '
112. Cylindrical, or Semi -polar Coordinates. If we take
the rectangular coordinate s, the perpendicular distance from the
Z-axis, Q7 and co the longitude, or angle made hy the plane includ-
ing the point M and the .Z'-axis, we have the system of semi-polar,
cylindrical, or columnar coordinates, for which we have immediately,
The parameter of a function f(z, p, o) is the resultant of the
partial parameters
= I ' p = + — - _ L
, ay
± '
113. Ellipsoidal Coordinates. Let us now find the value of
the parameter in terms of the ellipsoidal coordinates described in § 73,
which are defined for a point x, yy z as the three roots of the equation
1)
i y
"^ M
-1 = 0.
The three coordinate -surfaces at any point have been proved to be
mutually perpendicular at each point x, yy 2. Since the equation 1)
is an identity, we have, differentiating totally, that is changing x, y,8, 1,
zdz
Now if d^ is the perpendicular distance of the tangent plane from
the origin, we have by the last formula of § 110,
336 VIII. NEWTONIAN POTENTIAL FUNCTION.
A -i
so that we may write for the direction cosines of the normal,
COS
yd i
3) cos (my) =
cos
Now as we move along the normal, we have
*&i
dx = dn cos (**#) = ^2 , ^
*
dy =
^d;
6?^ = dn cos (WA^) = 2 .
C ~j~
Inserting these values in 2),
x* , 2/2
-|(^fiy2 + (^
so that
i
2 "
In order to express this result in terms of the elliptic coordinates
alone we may express x, y, 8, in terms of A, p, v. Observe that the
function
BY N _ 3? y* , z* -i
- -* + 2 + 2 -
has as roots A, ^, v, and being reduced to the common denominator
(e + O(e + &2)(9 + c2)
has a numerator of the third degree in Q. As this vanishes for
9 = 1, 9 = P> 9 = v,
it can only be
- (? - ).) (Q - p) (Q - v).
113] ELLIPSOIDAL COORDINATES. 337
Accordingly we have the identity
z\ F(o\ - x* , y" i =
- -1 r «" ~
Multiplying this by p + a2 and then putting p = — a2 we get
2 _ (aa + a)(aa + fO(*a + 'Q
r (a2-&2)(«2-c2)
and in like manner
o
* '
(&' -<>•)(&•-<»»)
_ .
If A, /i, v are contained in the intervals specified in § 73, these will
all be positive, so that the point will be real.
If we insert these values in d%, we shall have hi expressed in
terms of A, ^, v.
This is more easily accomplished as follows.
Differentiating the above identity 6) according to p,
- ft) (g - y) f 1 1
If we put p = A, all the terms on the right except the first, b.eing
multiplied by Q — A, vanish, and we have
, x*
22"1"
The expression on the left is — Therefore
In a similar manner we find
and the parameter of any function F(A, ^, v) is
WEBSTER, Dynamics. 22
338
VIII. NEWTONIAN POTENTIAL FUNCTION.
114. Infinitesimal Arc, Area and Volume. If we have any
three point -functions g1? g2, g3 forming an orthogonal system of co-
ordinates j since their parameters are
7 __ V lh ~L __ V */3
cn^ 2 dn^ 3 dns'
the normal distance between two
consecutive level surfaces q1 and
ql + dq± is dn± = -~i consequently
if we take six surfaces
&*%,&
Fig. 118.*
the edges of the infinitesimal curvi-
linear rectangular parallelepiped
whose edges are the intersections
of the surfaces are
and since the edges are mutually perpendicular, the diagonal, or
element of arc is
vrft#
"
dq3
the elements of area of the surfaces qlf q2, qs are respectively
and the element of volume is
Examples. Rectangular coordinates xf y, z.
Polar coordinates rf -9-, cp,
r sin -9-'
dSr = r^sin&d&dcp, element of area of sphere,
12) dS#=r sin#dr dcp, element of area of cone,
dS(p=rdrd& , element of area of plane,
dr =
114, 115] INFINITESIMAL SPACE -ELEMENTS.
Cylindrical coordinates, 0, 0, co,
339
13)
dSz = yd yd a, element of area of plane,
dSg = Qd&dz, element of area of cylinder,
(a = dQds , element of area of meridian plane,
dr
Elliptic coordinates, A, ^, v.
dpdv >/ Q -v}(\i- 1} (v -
2 + *0 (&1 + fQ (c2 + p) («2 + *) (fc2 + r) (c2+ v)
d»»
which will be referred to as Green's theorem in its second form.
We shall, unless the contrary is stated, always mean by n the
internal normal to a closed surface, but if necessary we shall
distinguish the normals drawn internally and externally as n{ and n*.
If we do not care to distinguish the inside from the outside we shall
denote the normals toward opposite sides by % and w2.
1) An Essay on the Application of Mathematical Analysis to the theories of
Electricity and Magnetism. Nottingham, 1828. Geo. Green, Reprint of papers, p. 25.
344 VIII. NEWTONIAN POTENTIAL FUNCTION.
116. Second Differential Parameter. If for the function U
we take a constant, say 1,
du du du
-fa~Ty== W= °> ^==0>
and we have simply
23) - jyprcos (Pvri) dS = fj d~^dS
The function
which, following the usage of the majority of writers, we shall denote
by z/F', was termed by Lame1) the second differential parameter of V.
As it is a scalar quantity it will be sufficiently distinguished from
the first parameter if we call it the scalar parameter. We have
accordingly the theorem giving the relation between the two: -
The volume integral of the scalar differential parameter of a
uniform continuous point -function throughout any volume is equal
to the surface integral of the vector parameter resolved along the
outward normal to the surface S bounding the volume.
We may obtain a geometrical notion of the significance of z/F
in a number of ways. In the neighborhood of a point 0, let us
develop V by Taylor's theorem, calling the coordinates of neighboring
points with respect to 0, xy y, z, then
where the suffix 0 denotes the value at 0.
Integrating the value of V -- F0 throughout the volume of a
small sphere with center at 0, we have
<£).///•*+! >
dr* s W "*" a2 """
therefore 4 V is equal to three times the mean of the second derivative
of V in a definite direction for all possible directions leading from
the point in question. This interpretation is due to Boussinesq.1)
By means of this result we may obtain a third interpretation
connecting the value of z/F at a point with the mean excess of
values on the surface of a small sphere, with center at the point,
over the value at the center.
If F0 denote the value at the center, the value at a distance R
in any direction is given by Taylor's theorem,
Integrating over the surface of a sphere of radius R, the deriva-
tives of V varying with the direction, since dS = R2d&, dividing by
the constant JR2
30) //(F- F0) da, = *ff
Now since
the terms in the first integral depend upon the directions simply
through the direction cosines of r, which on account of symmetry
cause the integral to vanish. If V is the mean of V on the surface
the equation then becomes
1) Boussinesq, Application des Potentiels a I' etude de Vequilibre et du mouve-
ment des solides elastiques, p. 45.
116, 117] DIVERGENCE THEOREM. 347
31) 4^(F-F0)
where the bar over the derivative denotes the mean for all directions
at 0, but this mean has just been proved to be equal to -^^JVai 0.
Consequently dividing by R2 and taking the limit the terms of higher
orders in E disappear and we have
32)
R=Q
The difference in the numerical coefficient in the two equations 27)
and 32) is accounted for by the fact that in 32) we have a mean
over a surface whereas in 27) we had a mean throughout a volume.
Any of the interpretations of the second differential parameter
shows that it is also a differential invariant. Thus Green's theorem
involves three different sorts of differential invariants.
117. Divergence. Solenoidal Vectors. If the components
of the vector parameter are
33)
Pcos(Ps) = £ = !?,
we have
OA\ *TT dX , 3Y . dZ
34) AV=^ — h -a — h -o->
dx " dy dz
and the theorem 23) becomes
35) - ffp cos (Pn) dS = -JJ[X cos (nx) + Fcos (ny)
If P is everywhere outward from the surface S, the integral is
positive, and
/dx . ar, ^^\^n
mean ( - — \- -5 — \~ -$— ) > 0.
\dx ' dy tiz J
Q -yr Q -T^- O ^
Accordingly -=— + ^ -- \- -^— is called the divergence of the vector
<7ic ' oy cz
point -function whose components are X; Y, Z, and will be denoted
by div. R.
The theorem as given in equation 35) may be stated as follows,
and will be referred to as the DIVERGENCE THEOREM: The mean value
348 VIII. NEWTONIAN POTENTIAL FUNCTION.
of the normal component of any vector point -function outward from
any closed surface S within which the function is uniform and con-
tinuous, multiplied ~by the area of the surface, is equal to the mean
value of the divergence of the vector in the space within S multiplied
ty its volume. The theorem is here proved for a vector which is the
parameter of a scalar point -function V, but it is evident that it may
be proved directly whether this is the case or not by putting in
equation 17) for W and x successively X, Y, Z and x, y, s respectively.
Let us consider the geometrical nature of a vector point -function E
whose divergence vanishes in a certain region. In the neighborhood
of any point, the vector will at some points be directed toward the
point and at others away. We may then draw curves of such a
nature that at every point of any curve the tangent is in the direc-
tion of the vector point -function E at that point. Such curves will
be called tines of the vector function. Their differential equations are
o£»\ dx dy dz
~X="Y~-~-~^'
Suppose that such lines be drawn through
all points of a closed curve, they will
generate a tubular surface, which will be
called a tube of the vector function. Let us
now construct any two surfaces Sl and 8.2
cutting across the vector tube and apply
the divergence theorem to the portion of
space inclosed by the tube and the two sur-
faces or caps S1 and S2. Since at every
point on the surface of the tube, E is
Fig. 123. tangent to the tube, the normal component
vanishes. The only parts contributing any-
thing to the surface integral are accordingly the caps, and since the
divergence everywhere vanishes in r, we have
37) C CE cos (En±) dS±+ f (*R cos (JR^) dS2 = 0.
SL S2
If we draw the normal to S2 in the other direction, so that as
we move the cap along the tube the direction of the normal is
continuous, the above formula becomes
38) (JE cos (En^ d8± = I I E cos (Rn^ dS2,
st s2
or the surface integral of the normal component of E over any cap
cutting the same vector tube is constant.
117] SOLENOIDAL VECTORS. 349
Such a vector will be termed solenoidal, or tubular, and the
O ~y Q -T7" O f7
condition - - + - + = 0 will be termed the solenoidal condition
ex cy oz
(Maxwell). We may abbreviate it, div. E = 0. If a vector point-
function JR is lamellar as well as solenoidal, the scalar function V
of which it is the vector parameter is harmonic, for
0X . dY . dZ
A solenoidal vector may be represented by its tubes, its direction
being given by the tangent to an infinitesimal tube, and its magnitude
being inversely proportional to its cross -section. As an example of
a solenoidal vector we may take the velocity of particles of a moving
liquid. If the velocity is B, with components X, Y, Z, the amount
of liquid flowing through an element of surface dS in unit time is
that contained in a prism of slant height E, and base dS, whose
volume is
E cos (En) dS.
The total flux, or quantity flowing in unit time through a sur-
face $, is the surface integral
I I EGOS (En) dS = I I [X cos (nx) -f Fcos (ny) -f Zcos (w*)] dS.
Such a surface integral may accordingly be called the flux of
the vector E through S.
A tube of the vector E is a tube through whose sides no fluid
flows, such as a material rigid tube through which a liquid flows,
and the divergence theorem shows that as much liquid flows in
through one cross -section as out through another, if the solenoidal
condition holds. If the liquid is incompressible, this must of course
be true.
As a second example of a solenoidal vector we have any vector
which is the curl of another vector, for
d (dZ 2Y\ _d_l^_<^2\ , JMU_^| =0
J^\dy~ dz\ + dy\ dz dx\^~dz\dx dy
identically.
The equation
is called Laplace's equation, and the operator
_a* d^ d^
~ dx* + dy*~^~ cz^
Laplace's operator.
350 VIII. NEWTONIAN POTENTIAL FUNCTION.
The parameter z/F is often called the Laplacian of V.
div. P = AV^ 0
Fig. 124 c.
div.P=AV = 0
Fig. 124 b.
In Fig. 124 a, b, c, are graphically
represented regions of divergent, solenoi-
dal, and convergent vector s, with the
level surfaces of the functions V of which
they are the vector parameters. The
arrows on the vector lines show the
direction of increase of V, and it is
evident that Fhas positive concentration
(and a maximum value) where P is
convergent, negative concentration (and
a minimum value) where P is divergent,
and no concentration (nor maximum)
where P is solenoidal.
118. Reciprocal Distance. Gauss's Theorem. Consider the
scalar point -function, F=— > where r is the distance from a fixed
point or pole 0. Then the level surfaces are spheres, and the para-
meter is
and since hr = 1,
- dr\r
drawn toward 0 (§ 110).
Consider the surface integral of the normal component of E
directed into the volume bounded by a closed surface S not con-
taining 0, or as we have called it, the flux of E into S,
40)
C CE cos (En) dS = - fC± cos (rn) dS.
117, 118]
GAUSS'S THEOREM.
351
The latter geometrical integral was reduced by Gauss. If to each
point in the boundary of an element dS we draw a radius and thus
get an infinitesimal cone with
vertex 0, and call the part
of the surface of a sphere of
radius r cut by this cone d2!,
dZl is the projection of dS
on the sphere, Fig. 125, and
as the normal to the sphere
is in the direction of r, we have
Fig. 125.
± dScos(rn),
the upper sign, for r cutting
in, the lower for r cutting
out. If now we draw about 0
a sphere of radius 1, whose area is 4#, and call the portion of its
area cut by the above-mentioned cone do, we have from the similarity
of the right sections of the cone
da>
The ratio d& is called the solid angle subtended by the infinitesimal cone.
Accordingly
dScoB (nr)
41)
. 7
= + - r — -+- do.
- r2
Now for every element dm, where r cuts into S, there is another
equal one, —do, where r cuts out, and the two annul each other.
Hence for 0 outside S,
42) ..
If on the contrary, 0 lies inside S, the integral I I dot is to be
taken over the whole of the unit sphere with the same sign, and
consequently gives the area 4 jr. Hence for 0 within S,
43)
rfcos(
JJ-1
These two results are known as Gauss's theorem, and the integral
will be called Gauss's integral.1)
1) Gauss, Theoria Attractions Corporum Sphaeroidicorum Ellipticorum
homogeneorum Methodo nova tractata. Werke, Bd. Y, p. 9.
352 VIII. NEWTONIAN POTENTIAL FUNCTION.
These results could have been obtained as direct results of the
divergence theorem. For the tubes of the vector function E are
cones with vertex 0. If 0 is outside 8, E is continuous in every
point within S, and since the area of any two spheres cut out by a
cone are proportional to the squares of the radii of the spheres, we
have the normal flux of
equal for all spherical caps. Consequently E is solenoidal, and the
flux through any closed surface is zero. If 0 is within S, E is
solenoidal in the space between 8 and any sphere with centre 0
lying entirely within S, and the flux through 8 is the same as the
flux through the sphere, which is evidently — 4 it.
The fact that E is solenoidal and V harmonic may be directly
shown by diiferentiation. If the coordinates of 0 are a, b, c,
44) r2 = (x - a)2 + (y - 6)2 + (* - c)2,
AK\ cr x — a Or y — b dr z — c
45) 7j- = -- 9 ~— = - - ? TT- = -- ;
ex r oy r oz r
—- _
dx\r r*dx~ rs '
_ _ I , 3 (a?-«) ll' = 3(a?-a)2-r8
f ~ f * ~ 5
8. (I) g.(I
48) ^ (i) = W +
\r/ dx* dyz
and — is harmonic, except where r = 0.
119. Definition and fundamental Properties of Potential.
We have seen in § 28, 34) that if we have any number of material
particles m repelling or attracting according to the Newtonian Law
of the inverse square of the distance, the function
= -y^ + ?p + ...+
'i '2
118, 119, 120] POTENTIAL FUNCTION. 353
where r19 r2, . . . rn are tlie distances from the repelling points, is the
force -function for all the forces acting upon the particle ms. If we
put the mass ms equal to unity, at a point P whose coordinates are
x, y, 2 the function
49) F=^ + ^ + . .+ ^=V-
n '» rn ^r
is called the potential function at the point P of the field of force
due to the actions of the particles mlt m2, ... mn, and y times its
negative vector parameter,
50) x = -rdJ, Y--rg, z--&,
f cx r dy * dz
is the strength of the field, that is, the force experienced hy unit
mass concentrated at the point xy y, 0.1)
Since any term — - possesses the same properties as the func-
i Tr
tion -9 § 118, we have for every term, for points where r is not
equal to zero, /4 \-\ = 0, and consequently
12O. Potential of Continuous Distribution. Suppose now
that the attracting masses, instead of being in discrete points, form
a continuously extended body K.
Let the limit of the ratio of the mass to the volume of any
infinitely small part be o = lim — •> which is called the density. Let
^r=0 4*
the coordinates of a point in the attracting body be a, &, c.
1) It is more usual among writers on attracting forces to write the force
as the positive parameter of the potential. The convention above adopted in 49)
amounts to defining the potential as the work necessary to remove the attracted
particle of unit mass from the given point to infinity against the attracting
forces, thus keeping the potential function positive, instead of negative as in
§ 28 (end). It is the usual practice to adopt such units that y is equal to
unity. In order to preserve consistency with the units previously employed and
at the same time not to be obliged to introduce y throughout all the equations
of this chapter, we shall define potential as above 49) and introduce the factor y
into those equations which involve the relationship of the force to the potential.
If the force is attractive, y will be negative, and putting y = — 1, we get the
usual formulae. Putting y = -|- 1, our notation agrees with that customary for
electricity and magnetism , for example in the author's Theory of Electricity and
Magnetism.
WEBSTER, Dynamics. 23
354
VIII. NEWTONIAN POTENTIAL FUNCTION.
The potential at any point P, x, y, s, due to the mass dm
at Q, a, 6, c, is
dF=^,
where r is the distance of
the point x, y, z from the
attracting point at a, b, c. The
whole potential at x, y, z is
the sum of that due to all
parts of the attracting body,
or the volume integral
Now we have
dm = ydr,
or in rectangular coordinates
rig. 126. dr^dadbdc,
dm = gdadbdc.
If the body is not homogeneous, p is different in different parts
of the body Ky and is a function of «, &, c, continuous or discon-
tinuous (e. g. a hole would cause a discontinuity). Since
53)
It is
For every point x, y, 2, V has a single, definite value,
accordingly a uniform function of the point P, x,y, 0.
It may be differentiated in any direction, we may find its level
surfaces, its first differential parameter, whose negative multiplied
by y is equal to the whole action of K on a point of unit mass,
and the lines of force, normal to the level, or equipotential surfaces.
If for any point x, y, z outside K, r± is the shortest distance to
any point of K, and r2 is the greatest distance, we have for any
point in K
dm dm
120, 121] POTENTIAL OF CONTINUOUS DISTRIBUTION. 355
Since r± and r2 are constant,
'
Now since / / / dm = M, the whole mass of the body K, the
K
above is
54) f '.,r-t
~s cos (rx) > - -^ cos (rx) > - -^ cos (r a?) .
rz - rt
Multiplying and dividing the outside terms by cos A and integrating,
Multiplying by jR2 and letting J5 increase without limit, since
,. JS2 ,. E* ,. cos(ra?)
lim -T = I™ — 5- = hm — = 1,
= -Jfcos^l,
62) lim [jR2 1?1 = - M cos ^,
Therefore the first derivatives of F, and hence the parameter,
vanish at infinity to the second order.
In like manner for the second derivatives,
a2F a« rcr^dr rrr v /i\ ,
^ = ^ J J J ~ = J J J ? w (7) dr
Every element in all the integrals discussed is finite, unless
r = 0, hence all the integrals are finite. We might proceed in this
manner, and should thus find that:
121, 122]
DERIVATIVES OF POTENTIAL.
357
At points not in the attracting masses, V and all its derivatives
are finite and (since their derivatives are finite) continuous, as well
as uniform.
Also since
63)
we have by addition
64)
f *j V y U fJ
that is, F satisfies Laplace's equation.
This is also proved by applying Gauss's theorem [§ 118, 42)] to
each element
r
122. Points in the Attracting Mass. Let us now examine
the potential and its derivatives at points in the substance of the
attracting mass.
If P is within the mass, the element — at which the point Q,
where dm is placed, coincides with P, becomes infinite. It does not
however, therefore follow that the integral
becomes infinite.
Let us separate from the mass K a
small sphere of radius s with the centre
at P. Call the part of the body within
this sphere K' and the rest K1'. Call
the part of the integral due to jfiT', Ff,
and that due to K", F". Now since P
is not in the mass K", F" and its deri-
vatives are finite at P, and we have only
to examine Ff and its derivatives.
Let us insert polar coordinates
Fig. 127.
000
so that, integrating first with respect to (p and #, since the absolute
value of an integral is never greater than the integral of the absolute
value of the integrand,
358 VIII. NEWTONIAN POTENTIAL FUNCTION.
65) \r |
if $m is the greatest value of Q in K'.
As we make the radius e diminish indefinitely this vanishes,
hence the limit
is finite.
In like manner for the derivative
d— =
dx~
Separate oft K' from X". The part of the integral from K"
is finite. In the other K' introduce polar coordinates, putting & = (rx),
66)
dV
dx
I dr I I \ sin -9- cos # | d&d -^- •
oy oz
32V
If we attempt this process for the second derivatives -~-^i • • •
it fails on account of — > which gives a logarithm becoming infinite
in the limit.
BV
We will give another proof of the finiteness of — • We have
which by Green's theorem is equal to
This is however only to be applied in case the function -- is
everywhere finite and continuous. This ceases to be the case when P
is in the attracting mass, hence we must exclude P by drawing a
122, 123] DERIVATIVES WITHIN MASS. 359
small sphere about it. Applying Green's theorem to the rest of the
space KUj we have to add to the surface -integral the integral over
the surface of the small sphere.
Since cos (nx) <^ 1, this is not greater than gm I I — = 4:iteQm,
which vanishes with £. Hence the infinite element of the integrand
contributes nothing to the integral.
In the same way that ^— was proved finite, it may be proved
dV 3V"
continuous. Dividing it into two parts -5— and -~ — > of which the
ox ox
second is continuous , we may make, as shown, -~ — as small as we
please by making the sphere at P small enough. At a neighbor-
ing point P! draw a small sphere, and let the corresponding parts
, 3V' , 3V" mi , 3V'
be -^ and -5-*— Then we can make ~^±- as small as we please,
ox ox ox
dV 3V'
and hence also the difference -~ --- -^L- Hence by taking P and
3V
P! near enough together, we can make the increment of ^- as small
o x
3V
as we please, or ^— is continuous, and accordingly the second derivatives
are finite.
123. Poisson's Equation. By Gauss's theorem [§ 118, 43)],
we have
when r is drawn from 0, a point within 8. Multiplying by m, a
mass concentrated at 0, and calling F=™>
68) ~ cos (>r) dS = -
The integral
n} ds>
where n is the internal normal, is the surface integral of the outivard
normal component of the parameter yP, or the inward component
of the force.
The surface integral of the normal component of force in the
inward direction through S is called the flux of force into 8, and
we see that it is equal to — 4#y times the element of mass within S.
Masses without contribute nothing to the integral. Every mass dm
360 VIII. NEWTONIAN POTENTIAL FUNCTION.
thin 8 contributes to the potential at
to the flux through the surface S. Therefore the whole
situated within 8 contributes to the potential at any point and
mass
; when the potential is F>= / / I -— > contributes to the flux
K
rrr
= " **rJJJ '
K
and
S K
Now the surface integral is, by the divergence theorem, equal to
70)
The surface 8 may be drawn inside the attracting mass, provid-
ing that we take for the potential only that due to matter in the
space V within S.
Accordingly for r we may take any part whatever of the attract-
ing mass, and
71)
As the above theorem applies to any field of integration what-
ever, we must have everywhere
72) z/F+ 4^0 = 0.
This is Poisson's extension of Laplace's equation, and says that
at any point the second differential parameter of F is equal to
- 4# times the density at that point. Outside the attracting bodies,
where Q = 0, this becomes Laplace's equation.
In our nomenclature, the concentration of the potential at any
point is proportional to the density at that point.
A more elementary proof of the same theorem may be given
as follows. At a point x,y,2 construct a small rectangular parallel-
epiped whose faces have the coordinates
x, x -f g, y, y -f vj, 0, z + 6,
and find the flux of force through its six faces. At the face normal
to the a;- axis whose x coordinate is x let the mean value of the
force be - d~ = - Px.
dx x
123] POISSON'S EQUATION. 301
The area of the face is 17 £, so that this face contributes to the
integral — / / Pco$(Pn)dS the amount — |i^g.
o i y-
At the opposite face, since o— is continuous, we have for its value
terms of hiher order in
and therefore, the normal being directed the other way, this side.
contributes to the integral the amount
and the two together
d*V
^y^Wx* ~^~ ^erms °^ higher order.
Similarly the faces perpendicular to Y-axis contribute o~
d*V
and the others %>n^ i [V
Thus the surface integral is
and by Gauss's theorem this is equal to
where Q is the mean density in the parallelepiped. Now making the
parallelepiped infinitely small, and dividing by |^g, we get
An important application of Poisson's equation has been made
to the attraction of the earth. The acceleration g is made up of the
resultant of the attraction of the earth and of the centrifugal accel-
eration. Since the latter has the components RPx, &?y along axes
perpendicular to the axis of rotation (§ 104), it has the potential
function — (x2 -f i/2) , so that if y denote the positive value of the
gravitation constant, and n the inward normal to an equipotential
surface, we have, putting
where
362 VIII. NEWTONIAN POTENTIAL FUNCTION.
is the potential of the earth's attraction. But by Poisson's equation,
so that we have,
74)
Now by the divergence theorem,
=-//!>.
so that
Now if the volume of the earth be v, its mean density gm, the volume
integrals are respectively equal to gmv and v, so that, multiplying by
this becomes
7Ax &2
76) ^m =
Thus if we know the value of g at every point on an equipotential
surface, we obtain the value of the product yqm in terms of the
angular velocity, and the surface integral of g. Using a formula
given by Helmert representing the results of geodetic determinations
of g, Woodward1) finds for the value of j>pm
= 3.6797 x 10
Richarz and Krigar-Menzel2) obtain, in a similar manner,
y$m = 3.680 x 10~ 7 see"2.
Combining this result with Boys's value of 7, p. 30 (see erratum), we
obtain for the mean density of the earth the value
124. Characteristics of Potential Function. We have
now found the following properties of the potential function.
1st. It is everywhere holomorphic, that is, uniform, finite, con-
tinuous.
1) Woodward, The Gravitational Constant and the Mean Density of the
Earth. Astronomical Journal, Jan. 1898.
2) F. Richarz und 0. Krigar-Menzel,, Gravitationsconstante und mittlere
Dichtigkeit der Erde, bestimmt durch Wagungen. Ann. der Phys. u. Chem. 36,
p. 177, 1898.
123, 124, 125] CHARACTERISTICS OF POTENTIAL. 363
2nd. Its first partial derivatives are everywhere holomorphic.
3rd. Its second derivatives are finite.
4th. V vanishes at infinity to the first order,
dV dV
' ~d~' ~dz vamsn *° second order,
lim(E2^} = -
R = K\ GX)
5th. F satisfies everywhere Poisson's differential equation
and outside of attracting matter, Laplace's equation
Any function having all these properties is a Newtonian potential
function.
The field of force X, Y, Z is a solenoidal vector at all points
outside of the attracting bodies, and hence if we construct tubes of
force, the flux of force is constant through any cross -section of a
given tube. A tube for which the flux is unity will be called a unit
tube. The conception of lines of force and of the solenoidal property
is due to Faraday.
Since F is a harmonic function outside of the attracting bodies,
it has neither maximum nor minimum in free space, but its maximum
and minimum must lie within the attracting bodies or at infinity.
In the attracting bodies the equation — z/F=47tp says that
the concentration of the potential at any point, or the divergence of
the force from it is proportional to the density at that point, except
where Q is discontinuous.
125. Examples. Potential of a homogeneous Sphere.
Let the radius of the sphere be B, h the distance of P from its
center,
Let us put s instead of r, using the latter symbol for the polar
coordinate,
364
YE!. NEWTONIAN POTENTIAL FUNCTION.
Now
Differentiating , keeping r
constant,
sds =
and introducing s as variable
instead of -9-,
Fig. 128.
If P is external we must integrate first with respect to s from
h — r to h r.
77)
0 A — r
Hence the attraction of a sphere upon an external point is the
same as if the whole mass were concentrated at the center.
A body having the property that the line of direction of its
resultant attraction on a point passes always through a fixed point
in the body is called centrobaric.
If instead of a whole sphere we consider a spherical shell of
internal radius JRt and outer R2, the limits for r being JR1? R2,
B,
78)
M
h'
We have
dV
M
dh*
h8
If, on the other hand, P is in the spherical cavity, h < JR17 the
limits for s are r — h, r -\- h
R2 r + h R.2
V= ^ CCrdrds = 4=*$ Crdr
79)
which is independent of fc, that is, is constant in the whole cavity.
/} V
Hence = 0, and we get the theorem due to Newton that a homo-
125]
ATTRACTION OF SPHERE.
365
geneous spherical shell exercises no force on a body within. (On
account of symmetry the force can be only radial.)
If P is in the substance of the shell, we divide the shell into
two by a concentric spherical surface passing through P, find the
potential due to the part within P, and add it to that without, getting
80) F =
8A
dV
dh ~ 3 h*
dh*~ 3 hs
Tabulating these results,
81)
F
dV
dh
3
3 h*
Plotting the above results (Fig. 129) shows the continuity of V
and its first derivative and the discontinuity of the second derivative
at the surfaces of the attracting
mass.
.We see that the attraction
of a solid sphere at a point
within it is proportional to the
distance from the center, for if
fe-o,
dV
and is independent of the radius
of the sphere. Hence experi-
ments on the decrease of the
force of gravity in mines at
known depths might give us
the dimensions of the earth, if
the earth were homogeneous. Experiment shows, however, that. this
is not the case.
- 129-
366
VIIL NEWTONIAN POTENTIAL FUNCTION.
126. Disc, Cylinder, Cone. Let us find the attraction of a
circular disc of infinitesimal thickness at a point on a line normal
to the disc at its center. Let the radius be R, thickness a, distance
of P from the center In.
0 0
= 2xeQ{yh2 + R2-h],
Attraction of circular cylinder on
point in its axis. Let the length be I
and let the point be external, at a
distance Ji from the center.
By the above, a disc of thickness
dx at a distance x from the center
produces a potential at P
-x}2 -(h- x)}.
Hence the whole is
84) V-
Circular cone on point in axis.
126, 127]
DISC, CYLINDER, CONE.
367
Let E be the radius of base, a the altitude, h the height of P
above the vertex.
A disc at distance x below vertex and radius r causes potential
at P,
and
E
E
85)
If we have a conical mountain of uniform , density on the earth,
and determine the force of gravity at its summit and at the sea level,
this gives us the ratio of the attraction
of the sphere and cone to that of the
sphere alone, and from this we get
the ratio of the mass of the earth to
the mass of the mountain. Such a
determination was carried out by
Mendenhall, on Fujiyama, Japan, in
1880, giving 5.77 for the earth's
density. m* 1S1-
Circular disc on point not on axis. Let the coordinates of P
with respect to the center be a, &, 0. Then
s2 = a2 -f (b — r cos
yt 6.
This is Poisson's equation for a surface distribution. If we draw
the normal away from the surface on each side, we may write
94)
or
l cos t
WEBSTER, Dynamics.
dV , dV
•K h o — = •—
r
cos
24
370
VIII. NEWTONIAN POTENTIAL FUNCTION.
128. Green's Formulae. Let us apply Green's theorem to
two functions, of which one, V, is the potential function due to
any distribution of matter, and the other, U = — > where r is the
distance from a fixed point P, lying in the space x over which we
take the integral. Let the space t
concerned he that hounded hy a
closed surface S, a small sphere H
of radius s about P, and, if P is
without S, a sphere of infinite radius
with center P.
Now the theorem was stated in
§ 115; 22) for the normal drawn in
toward T, which means outward from S and 27, and inward from
the infinite sphere, as
. 184.
and since
in the whole space r, so that 1) becomes
The surface integrals are to be taken over S, over the small sphere,
and over the infinite sphere. For a sphere with center at P,
-5— = - -5— = __ — ,
dn ~ dr r r*
the upper or lower sign being taken according as the sphere is the
inner or outer boundary of T;
and for
r = oo
V vanishes, hence this integral vanishes. Also
O T7" j
Now at infinity, -- is of order -# and being multiplied by r, still
128] GREEN'S FORMULAE. 371
vanishes. Accordingly the infinite sphere contributes nothing. For
the small sphere the case is different. The first integral
- / / VdG)
*J *J
becomes, as the radius s of the sphere diminishes,
4) -^//*—
The second part
however, since -^ is finite in the sphere, vanishes with s. Hence
there remain on the left side of the equation only — 4:7tVP and the
integral over 8. We obtain therefore
the normal being drawn outward from 8. This formula is due to
Green.
Therefore we see that any function which is uniform and con-
tinuous everywhere outside of a certain closed surface, which vanishes
at infinity to the first order, and whose parameter vanishes at infinity
to the second order, is determined at every point of space considered
if we know at every point of that space the value of the second
differential parameter, and in addition the values on the surface 8 of
the function and its vector parameter resolved in the direction of
the outer normal.
In particular, if V is harmonic in all the space considered, we have
K\ V
b) . VP — — -
'
and a harmonic function is determined everywhere by its values and
those of its normal component of parameter at all points of the
surface 8.
Since
r
== - i | cos (nx) || + cos (ny) ~ry + cos (ne) fz } =
cos(nr)
a f
24
372 VIII. NEWTONIAN POTENTIAL FUNCTION.
we may write 6)
7) Vp=~A
Consequently, we may produce at all points outside of a closed
surface S the same field of force as is produced by any distribution
of masses lying inside of S, whose potential is F, if we distribute
over the surface S a surface distribution of surface -density,
1 (Fcos(wr) d_V\
8) 6 =
In the general expression, 5) the surface integral represents the
potential due to the masses within S, while the volume integral
-mm
since everywhere
is equal to
that is, the potential due to all the masses in the region T, viz.,
outside S.
129. Equipotential Layers. As a still more particular case
of 7), if the surface S is taken as one of the equipotential surfaces
of the internal distribution, we have all over the surface V= Vs — const.,
and the constant may be taken out from the first integral,
9) Fp=_ d8_
±*J J r ^JJ r $n
Now by Gauss's theorem / / C^f^dS = Q, accordingly,
so that VP is represented as the potential of a surface distribution
of surface -density
l dV IF ,-v N +1F
G = — - - -5-- = - - cos (Fn) = ~f ---
4:7C on 4:7f y kit y
The whole mass of the equivalent surface distribution is
128, 129] EQUIPOTENTIAL LAYERS. 373
which, heing the flux of force outward from S, is by Gauss's theorem,
§ 123, 68), equal to M, the mass within 8.
Accordingly we may enunciate the theorem, due to Chasles and
Gauss1): —
We may produce outside any equipotential surface of a distribu-
tion M the same effect as the distribution itself produces, by dis-
tributing over that surface a layer of surface -density equal to -
times the outward force at every point of the surface. The mass of
the whole layer will be precisely that of the original internal dis-
tribution. Such a layer is called an equipotential layer. (Definition
— A superficial layer which coincides with one of its own equi-
potential surfaces.) Reversing the sign of this density will give us
a layer which will, outside, neutralize the effect of the bodies within.
The above theorem has an important application in determining
the attraction of the earth at outside points. Equation 10) shows that
the potential and therefore the attraction is determined at all outside
points if F, which is connected with g as in § 123, is known at all
points of an equipotential surface. It will be shown later that the
surface of the sea is an equipotential surface. Consequently if the
value of g is known from pendulum observations at a sufficient
number of stations distributed over the surface of the earth the
attraction at all points outside the earth can be calculated.
Let us now suppose the point P is within S. In this case, we
apply Green's theorem to the space within S, and we do not have
the integrals over the infinite sphere. The normal in the above
formulae is now drawn inward from S, or if we still wish to use the
outward normal, we change the sign of the surface integral in 5),
12) vp = - r _ I T A8 _ _L
r
(P inside 8).
Note that both formulae 5) are 12) are identical if the normal
is drawn into the space in which P lies.
Hence within a closed surface a holomorphic function is deter-
mined at every point solely by its values and those of its normally
resolved parameter at all points of the surface, and by the values of
its second parameter at all points in the space within the surface.
A harmonic function may be represented by a potential function
of a surface distribution.
1) Chasles, Sur I' attraction d'une couche ellipsoidale infiniment mince, Journ.
. Polytec., Cahier 25, p. 266, 1837; Gauss, AUgemeine Lehrsatze, § 36.
374 VIII. NEWTONIAN POTENTIAL FUNCTION.
Now if the surface S is equip otential, the function V cannot be
harmonic everywhere within unless it is constant. For since two
equipotential surfaces cannot cut each other, the potential being a
one -valued function, successive equipotential surfaces must surround
each other, and the innermost one, which is reduced to a point, will
be a point of maximum or minimum. But we have seen (§ 116)
that this is impossible for a harmonic function. Accordingly a func-
tion constant on a closed surface and harmonic within must be a
constant.
If however there be matter within and without S, the volume
integral, as before, denotes the potential due to the matter in the
space" r (within 8), and the surface integral that due to the matter
without. If the surface is equipotential, the surface integral
&£/#&«
**, ±*J J r dne
(*^-dS + ~
The first integral is now equal to 4jr, so that
Vs being constant contributes nothing to the derivatives of F, so that
the outside bodies may be replaced by a surface layer of density
14) tf = - = _ .Pcos (Fne) = +
4tt dng kny -
The mass of the surface distribution,
15)
ne being the outward normal, is the inward flux of force through S,
which is equal to minus the mass of the interim1 matter, and is not,
as in the former case, equal to the mass which it replaces.
13O. Gauss's Mean Theorem. As an example of equation 6)
let us make the surface S a sphere with center at P. Then in the
first term of the integral we have
which is constant and may be taken outside the integral. In the
second term J being similarly taken outside the integral, we have
129, 130, 131]
GAUSS'S MEAN THEOREM.
375
since the function is harmonic in the sphere considered. Accordingly
the formula reduces to the first term
16)
VdS.
The surface integral represents the mean value of F on the surface
of the sphere multiplied by the area of the surface, 4^r2. Thus we
have the theorem due to Gauss. The value of the potential at any
point not situated in attracting matter is equal to the mean value of
the potential at points on any sphere with center at the given point
and not containing attracting matter. It at once follows from this
theorem that a harmonic function cannot have a point of maximum
or minimum, for making the sphere about such a point small enough
the theorem would be violated.
131. Potential completely determined by its charac-
teristic Properties. We have proved that the potential function
due to any volume distribution has the following properties:
1. It is, together with its first differential parameter, uniform,
finite, and continuous.
2. It vanishes to the first order at oo, and its parameter to the
second order.
3. It is harmonic outside the attracting bodies, and in them
satisfies
The preceding investiga-
tion shows that a function
having these properties is a
potential function, and is
completely determined.
For we may apply the
above formula 5) to all space,
and then the only surface
integral being that due to
the infinite sphere, which
vanishes, we have
Fig. 135.
If however, the above conditions are fulfilled by a function F,
except that a certain surfaces S its first parameter is discontinuous,
376 VIII. NEWTONIAN POTENTIAL FUNCTION.
let us draw on each side of the surface S surfaces at distances
equal to e from S, and exclude that portion of space lying between
these, which we will call S± and $2.
If the normals are drawn into r we have
•> '-
The surface integrals are to be taken over both surfaces S1 and
$2 and the volume integrals over all space except the thin layer
between S1 and 82. This is the only region where there is discon-
tinuity, hence in x the theorem applies, and
is, 4,r-j£ (i) «, +j r£ (1)
Now let us make e infinitesimal, then the surfaces 8lf $2 approach
each other and 8. V is continuous at 8, that is, is the same on
both sides, hence, since = — (—) = — - — (— ), in the limit the first
3*V\r/ dn^\rj
two terms destroy each other. This is not so for the next two.
3 V 7}V
for ~— is not equal to -= — because of the discontinuity.
In the limit, then
The volume integral, as before, denotes the potential / / / -: dr
due to the volume distribution, while the surface integral denotes
the potential of a surface distribution / / - > where
dV
-
. Hence we get a new proof of Poisson's surface condition,
§ 127, 94).
132. Kelvin and Dirichlet's Principle. We shall now
consider a question known on the continent of Europe as Dirichlet's
Problem.
Given the values of a function at all points of a closed surface
8 - - is it possible to find a function which, assuming these values
on the surface, is, with its parameters, uniform, finite, continuous,
and is itself harmonic at all points within 8?
131, 132] PRINCIPLE OF KELVIN -DIRICHLET. 377
This is the internal problem — the external may be stated in
like manner, specifying the conditions as to vanishing at infinity.
Consider the integral
of a function u throughout the space r within 8.
J must be positive, for every element is a sum of squares. It
cannot vanish, unless everywhere -5- = -^ = -= - = 0. that is u = const.
dx dy dz
But since u is continuous, unless it is constant on S, this will not
be the case.
Accordingly J(M) > Q
Now of the infinite variety of functions u there must be,
according to Dirichlet, at least one which makes J less than for
any of the others. Call this function v, and call the difference
between this and any other hs, so that
u = v -+• hs,
h being constant.
The condition for a minimum is that
J(v) A\ 72 TV N 07 C C Ci^V $S , ^V ds 'd® B 8\ -j n
24) WJ(s} + M- + -r + 3sdr' °'
for aZZ values of ft, positive or negative. But as s is as yet un-
limited, we may take h so small that the absolute value of the term
in li is greater than that of the term in h2, and if we choose the
378 VIII. NEWTONIAN POTENTIAL FUNCTION.
sign of Ji opposite to that of the integral, making the product
negative, the whole will be negative.
The only way to leave the sum always positive is to have the
integral vanish. (It will he observed that the above is exactly the
process of the calculus of variations. We might put 6 v instead of hs.}
The condition for a minimum is then
But by Green's theorem, this is equal to
Now at the surface the function is given, hence n and v must
have the same values, and s = 0.
Consequently the surface integral vanishes, and
= 0.
But since s is arbitrary, the integral can vanish only if everywhere
in T, z/v = 0, v is therefore the function which solves the problem.
The proof of the so-called Existence -theorem, namely, that there
is such a function, depends on the assumption that there is a function
which makes the integral J a minimum. This assumption has been
declared by Weierstrass, Kronecker, and others, to be faulty. The
principle of Lord Kelvin and Dirichlet, which declares that there is
a function v, has been rigidly proved for a number of special cases,
but the above general proof is no longer admitted. It is given here
on account of its historical interest.1)
We can however prove that if there is a function v, satisfying
the conditions, it is unique. For, if there is another, v\ put
1) Thomson, Theorems with reference to the solution of certain Partial
Differential Equations, Cambridge and Dublin Math. Journ., Jan. 1848; Reprint
of Papers in Electrostatics and Magnetism, XIII. The name Dirichlet' s Prinzip
was given by Riemann (Werke, p. 90). For a historical and critical discussion
of this matter the student may consult Burkhardt, Potentialtheorie in the
Encyklopadie der Mathematik, Bacharach, Abriss der Greschichte der Potential-
theorie, as well as Harkness and Morley, Theory of Functions, Chap. IX, Picard,
Traite d' Analyse, Tom. II, p. 38. It has been quite recently shown by Hilbert
that Riemann's proof given above can be so modified as to be made rigid.
132, 133] GREEN'S THEOREM IN CURVILINEAR COORDINATES.
379
On the surface, since
0. In T, since
and
are zero, 4u = 0. Accordingly J(u) = 0. But, as we have shown,
this can only be if u = const. But on S, u = 0, hence, throughout
r9 u = 0 and v = v1.
133. Green's Theorem in Orthogonal Curvilinear Co-
ordinates. We shall now consider Green's theorem in terms of
any orthogonal coordinates, beginning with the special case forming
the divergence theorem, § 117, 35).
28) -if [X cos (nx) + Tcos
+ Zco8(n#y\
~/ff13-
dY
dz\,
^-\dr.
Instead of the components X, Y, Z, let us consider the projec-
tions P1; P2, P3 of a vector P along the directions of the tangents
to the coordinate lines q_uq%,q% at any point. Then projecting along
the normal n to S, we have the integrand in the surface integral
29)
If we divide the volume T
up into elementary curved
prisms bounded by level sur-
faces of q2 and g3, as in the
case of rectangular coordinates
(Fig. 136), we have, at each
case of cutting into or out
of S respectively,
Pt cos (nn±) -f- P2 cos (nn2) + P3 cos
dS
{,
± dScos(nni)
where dS^ is the area of the
part out by the prism from
the level surface qv
Now by § 114,
~ dqz dqs
accordingly
30) -
Fig. 136.
the change from the double to the triple integral involving the same
considerations as in the proof given for rectangular coordinates in
§ 115.
380 vm- NEWTONIAN POTENTIAL FUNCTION.
Transforming the other two integrals in like manner,
L cos (nn±) -f P2 cos (nn2) + P3 cos (
_ / / {
But this is equal to
But since
multiplying and dividing the last integrand in 31) by \\li^ we find
that since the volume integrals are equal for any volume, the integrands
must he equal, or
If the vector is lamellar, its projections are the partial parameters
according to &, &, & of its potential V (§ 110),
Pi = hi /S - ? Pj> = fas) Q - f P a = lln Q - '
18fX 2^?2 3^?8
Equation 32) then becomes
33)
This result for the value of z/F was given by Lame, by means
of a laborious direct transformation. The method here used is a
modification of one given by Jacobi and Somoff.1)
In order to prove Green's theorem in its general form, we remark
that from the nature of the mixed parameter of the two functions U
and F as a geometric product we have
34) A ( U, F) = PfPf + PfP2F + Pf P3r
,, dU dV . 19 dU dV , 72 dU dV
= M o— « — h /*; o^ y — h ">\
Forming the volume integral, and integrating the first term partially
according to q± we obtain
1) Lame, Journal de I'Ecole Poly technique , Cahier 23, p. 215, 1833; Legons
sur les Coordonnees curvilignes, II. Jacobi, Uber eine particular e Losung der
partiellen Differentialgleichung JV=Q, Crelle's Journal, Bd. 36, p. 113. Somoff,
Theoretische Mechanik, II. Teil, §§ 51, 52.
133, 134] STOKES'S THEOREM IN CURVILINEAR COORDINATES. 381
35)
'
which as above is equal to
Integrating the other terms in like manner we obtain the general
formula,
Q*\ CCC\i*dUdV , J93UdV . 19 dU dV\ dq.dq.dq,
36) / / / {W*— js -- h^lo— 5 --- h ---—
y ' 8
= -J J Z?
in which each integrand is found to correspond to one of those in
§ 115, 20).
134. Stokes's Theorem in Orthogonal Curvilinear Co-
ordinates. The proof of Stokes's theorem given in § 30 can be
easily adapted to curvilinear coordinates.1) Let Pt, P2; P3 be the
projections of a vector P on the varying directions of the tangents
to the coordinate lines at any point. Then, the projections of the
arc ds being ds19 ds%, dss, we consider the line -integral
37) I =p cos (P, ds) ds = i dsi + P2 ^ 4- P
A
B
where
1) Webster. Note on Stokes's theorem in Curvilinear Coordinates. Bull.
Am. Math. Soc., 2nd Ser., Vol. IV., p. 438, 1898.
382 VIII. NEWTONIAN POTENTIAL FUNCTION.
Let us now make an infinitesimal transformation of the curve
as in § 30. Then the change in the integral is
B
38) dI==
The last three terms can be integrated by parts, giving
B B
39) f Esddqs = Esdqs - fiqsdEs, (* = 1,2,3),
A A
and, since the integrated terms vanish at the limits,
40) dI=J(dE1dq±+ dE, dq2 + dE^ dqz -dE1 dq,- dE, dq2- dE3 d &).
Performing the operations denoted by d and d, as on p. 85, six of
the eighteen terms cancel, and there remain the terms,
Now the changes dg9, dq%, d#3, dqs, in the coordinates correspond
to distances , ,
measured along the coordinate lines, and the determinant of these
distances,
is equal to the area of the projection on the surface q± of the
infinitesimal parallelogram swept over by the arc ds during the trans-
formation. Calling this area dS, and its normal n, we have
dqs - dqB dq^) = cos 0%) dS.
If we now continually repeat the transformation, until the curve 1
joining AB is transformed into the curve 2, the total change in 1
is equal to the surface integral over the intervening surface,
42)
134, 135] LAPLACE'S EQUATION IN SPHERICAL COORDINATES. 383
Accordingly the components of (7
== -- = — > as before.
Q Q
If we had attempted to verify the value 48) of V by direct
calculation, we should have found a difficulty in the appearance of
a logarithm which would have become infinite when the length of
the cylinder became infinite. Nevertheless the attraction is finite, as
just shown. It is to be note that all the properties hitherto proved
to hold have been for potentials of bodies of finite extent.
136. Logarithmic Potential. We may state the above result
in terms of the following two-dimensional problem. Suppose that
on a plane there be distributed a layer of mass in such a way that
a point of mass m repels a point of unit mass in the plane with a
force — where r is their distance apart. The potential due to m is
V = — m log r and it satisfies the differential equation
a«r a»r 0
+ =
Similarly, in the case of any mass distributed in the plane, with
surface -density /it, an element dm = [jidS produces the potential
, and the whole the potential
51) F = -r f fdwlogr = - I CplogrdS,
where r is the distance from the repelled point x, y to the repelling
dm at a, &, so that
We may verify by direct differentiation that, at external points,
this V satisfies
dv
dx
== ~~ WxJ J
WEBSTER, Dynamics. 25
386 VIII. NEWTONIAN POTENTIAL FUNCTION.
2(i/-&)2
This potential is called the logarithmic potential and is of great
importance in the theory of functions of a complex variable.
137. Green's Theorem for a Plane. In exactly the same
manner that we proved Green's Theorem for three dimensions , we
may prove it when the integral is the double integral in a plane
over an area A bounded by any closed contour C. Since we have
for a continuous function TF
53) dxdy =[W, -W, +• • . + TF2w - W2n^] dy
= — I TFcos (nx) dSj
cj
where n is the inward normal, ds the element of arc of the contour.
dV
Applying this to W=U-^--y we obtain
54)
Treating the other term in like manner, we obtain
C A
Interchanging U and V we obtain the second form
ds= I I (V4U— U4V)dxdy,
c A
where we write ^y
'•-^
136, 137, 138]
LOGARITHMIC POTENTIAL.
337
138. Application to Logarithmic Potential. If in 56)
we put U= 1, we obtain
57)
which is the divergence theorem in two
dimensions. If the function V is harmonic
everywhere within the contour, we have
0-
Applying this to the harmonic func-
tion logr, where P, the fixed pole from
which r is measured, is outside the contour,
Cdlogr -, I l..r 4 /*cos(rw)
/ —5^- ds = I - TT- ds = / - —
J dn J r dn J r
If the pole P is within the contour, we draw a circle K of any
radius about the pole, and apply the theorem to the area outside of
this circle and within the contour, obtaining the sum of the integrals
around C and K equal to zero, or
rf\\
59)
cos (rri)
These two results are Gauss's theorem for two dimensions. They
may of course be deduced geometrically in the same way as for three
dimensions, § 118. We may now deduce Poisson's equation for the
logarithmic potential as in § 123 for the Newtonian Potential. The
logarithmic potential due to a mass dm being — dmlogr gives rise
to the flux of force 2ttdm outward through any closed contour
surrounding it, and a total mass m causes the flux
= 2it / /
pdxdy.
Put in terms of the potential this is
60) Cj^ds = - f f^Vdxdy = 2%
C A A
and since this is true for any area of the plane, we must have
61) JV=-2itii.
This is Poisson's equation for the logarithmic potential.
25*
388 VIII. NEWTONIAN POTENTIAL FUNCTION.
139. Green's Formula for Logarithmic Potential. Apply-
ing Green's Theorem 56) to the functions logr and any harmonic
function V, supposing the pole of P to be within the contour, and
extending the integral to the area within the contour and without a
circle K of radius £ about the pole,
c
The third term is
(since V is harmonic in K) and the fourth,
_ (Y^ds = - f^rd^ = - Cvd»,
J dn J r J
K K K
which, when we make e decrease indefinitely, becomes
Accordingly we obtain the equation
63) . F,
which is the analogue of equation 6), § 128. In a similar way we
may find for nearly every theorem on the Newtonian Potential a
corresponding theorem for the Logarithmic Potential. A comparison
of the corresponding theorems will be found in C. Neumann's work,
Untersuchungen uber das logarithmische und das Newtonsche Potential.1)
The Kelyin-Dirichlet Problem and Principle may be stated and
demonstrated for the logarithmic potential precisely as in § 132.
14O. Dirichlet's Problem for a Circle. Trigonometric
Series. We shall call a homogeneous harmonic function of order n
of the coordinates x, y of a point in a plane a Circular Harmonic,
since it is equal to $n multiplied by a homogeneous function of
cos G) and sin 09, and consequently on the circumference of a circle
about the origin is simply a trigonometric function of the angular
coordinate ra. Any homogeneous function Vn of degree n satisfies
the differential equation
1) See also Harnack, Die Grundlagen der Theorie des logarithmischen Poten-
tiates; Picard, Traite d} Analyse, torn. II; Poincare, Theorie du Potentiel Newtomen.
139, 140] TRIGONOMETRIC SERIES. 389
so that a circular harmonic is a solution of this and Laplace's Equa-
tion simultaneously. The homogeneous function of degree n
anxn + dn-ix"-1 H a^xyn~l -f a • yn
contains n + 1 terms, the sum of its second derivatives is a homo-
geneous function of degree n — 2 containing n — 1 terms, and if this
is to vanish identically each of its n — 1 coefficients must vanish,
consequently there are n — 1 relations between the n + 1 coefficients
of Vny or only two are arbitrary. Accordingly all harmonics of
degree n can be expressed in terms of two independent ones. The
theory of functions of a complex variable1) tells us that the real
functions u(x, y), v(x, y) in the complex variable u -f- ivf which is a
function of the complex variable x -\- iy, are harmonic functions of
x, y, and making use of Euler's fundamental formula,
65) x -f iy = Q {cos co -f i sin co} = geita,
and raising to the nih power, we have
66) (x 4- iy)n = Qnein(a = $n (cos no + i sin no).
Accordingly we have the two typical harmonic functions
£» rr\ « w *
It may be at once shown that these functions are harmonic by
substitution in Laplace's equation in polar coordinates, equation 47).
Accordingly the general harmonic of degree n is
/^ Q \ T 7" 47 ( A | TO * \ m /TT
We may call the trigonometric factor Tn, which is the value of the
harmonic on the circumference of a circle of radius unity, the
peripheral harmonic of degree n.
If a function which is harmonic in a circular area can l>e
developed in an infinite trigonometric series
69) V(x, y} ••
on the circumference of the circle of radius R, the solution of
Dirichlet's Problem for the interior of the circle is given by the series
For every term is harmonic, and therefore the series, if convergent,
is harmonic. At the circumference Q = JR, and the series takes the
given values of V. The absolute value of every term is less than
the absolute value of the corresponding term in the series 69), in
1) See § 197.
390
VIII. NEWTONIAN POTENTIAL FUNCTION.
virtue of the factor — ? therefore if the series 69) converges, the
-pn f
series 70) does as well. Since the series fulfils all conditions, by
Dirichlet's principle it is the only function satisfying them.
We may fulfil the outer problem by means of harmonics of
negative degree. Taking n negative , the series
71)
RT\
Q
is convergent , takes the required values on the circumference, and
vanishes at infinity. For a ring-shaped area between two concentric
circles, we may satisfy the conditions by a series in both positive
and negative harmonics,
72)
H- ^? Q~*n{A!n cos no
140 a. Development in Circular Harmonics. We may use
the formula 63), § 139, to obtain the development of a function in
a trigonometric series on the circumference of
a circle. Let the polar coordinates of a point
on the circumference of the circle be B, o and
of a point P within the circumference 0, cp.
Then we have for the distance between the
two points
rig. 139.
Removing the factor R2, inserting for cos (CD — Q)
its value in exponentials, and separating into
factors we obtain
73)
r = E
Taking the logarithm we may develop
and
140, 140 aj DEVELOPMENT IN TRIGONOMETRIC SERIES. 391
by Taylor's Theorem, obtaining
74) logr = logE - 45?- — (en*<»-y> + e-«»-(«-9))
= log B -~e~ cos » (o> - ?).
1
This series is convergent if Q < E, and also if Q = E, unless
03 = (p.
Inserting this value of logr in 63), differentiation with respect
to the normal being according to —E, we have
75)
Expanding the cosines, we may take out from each term of the
integral, except the first, a factor Qncosncp or pnsinwgp, so that VP
is developed as a function of its coordinates p, cp, in an infinite series
of circular harmonics, the coefficients of which are definite integrals
around the circumference, involving the peripheral values of V
and •*— This does not establish the convergence of the series on
the circumference. Admitting the possibility of the development, we
may proceed to find it in a more convenient form. In order to do
this let us apply the last equation to a function Vm, which is a
circular harmonic of degree m. Then at the circumference we have
T7" ~Dm T> m -r>m — 1 rrj
Vm = ±i JLm, -Q^ = -witf JLm,
and
271
76) Fm(P) = ^*-
The expression on the right is an infinite series in powers of p,
while Vm(P) is simply QmTm. As this equality must hold for all
392 vnl- NEWTONIAN POTENTIAL FUNCTION.
values of Q less than JR, the coefficient of every power of Q except
the mih must vanish, and we have the important equations
77) / Tmcosn((o — (p)do = 0,
o
78) Tm() = V (An cos n o + Bn sin n o) = V Tn (o).
0 0
Multiply both sides by cosm (a — (p)d& and integrate from 0 to 2 it.
81) / F(o) cos m (03 — (p)dc3=^. I Tn((ai) cos m(a) — w}do.
*J ^^ *J
o oo
Every term on the right vanishes except the mth which is equal to
nTm((p). Accordingly we find for the circular harmonic Tm the
definite integral Sft
82) Tm(y) = — I F(to) cos m (o — cp) do.
o
For m = 0, we must divide by 2.
Writing for Tm(jtp) its value
Am cos mcp + Bm sin wqp,
expanding the cosine in the integral, and writing the two terms
separately, we obtain the coefficients
83) AQ = ~ I F(ra) d&j Am = — I F(o) cos mco do,
o o
27T
Bm = — I ^C03) sinmodco.
140 a, 141] SPHERICAL HARMONICS. 393
This form for the coefficients was given by Fourier1), who assuming
that the development was possible, was able to determine the coefficients.
The question of proving that the development thus found actually
represents the function, and the determination of the conditions that
the development shall be possible, formed one of the most important
mathematical questions of this century, which was first satisfactorily
treated by Dirichlet.2) For the full and rigid treatment of this
important subject, the student should consult Kronecker, Theorie der
einfachen und der vielfachen Integrate; Picard, Traite d 'Analyse, Tom. I,
Chap. IX; Riemann-Weber, Partielle Differentialgleichungen ; Poincare,
Theorie du Potentiel Newtonien.9)
141. Spherical Harmonics. A Spherical Harmonic of degree n
is defined as a homogeneous harmonic function of the coordinates
x, y, z of a point in space, that is as a solution of the simultaneous
equations
84)
Q*\ ° i ,
85) x -a— + V -o-H- ^ -Q— = n V.
dx v dy dz
The general homogeneous function of degree n
anQxn + an^lt0xn-1y + an-2,<>xn~2y2 h
-f an-Litf*-^ + an-2,ixn-2y^ f-
contains 1 + 2 + 3 • • • + n + 1 = terms> ^ gum of itg
second derivatives is a homogeneous function of degree n — 2 and
accordingly contains 2 'n terms. If the function is to vanish
identically, these 2 l coefficients must all vanish, so that there
(n— 1) n , .. n (n 4- 1) (n 4- 2) „„ . „ ,
are - — ~— relations among the - ^ - coefficients of a harmonic
of the nih degree, leaving 2^ + 1 arbitrary coefficients. The general
harmonic of degree n can accordingly be expressed as a linear func-
tion of 2n + 1 independent harmonics.
1) Fourier, Theorie analytique de la Chaleur, Chap. IX, 1822.
2) Dirichlet, "Sur la Convergence des Series Trigonometriques1', Crelle's
Journal, Bd. 4, 1829.
3) A resume of the literature is given by Sachse, Bulletin des Sciences
MatMmatiques , 1880.
394 VIII. NEWTONIAN POTENTIAL FUNCTION.
Examples. Differentiating the arbitrary homogeneous function,
and determining the coefficients, we find for n = 0, 1, 27 3, the
following independent harmonics:
n = 0 a constant,
n = l x, y, e,
n = 2 x2 — y2, f — £, xy, yz, zx,
n = 3 3x*y-y3, 3x22-z3, 3y2x-x*, 3y*0-0*, 3z2x-x3, 3*2y-y8, xyz.
If we insert spherical coordinates r, ft, a £)/? cy
0 JL ° y
I ^ « /? ^ v ' nJ
is a harmonic of degree »-(**£+ ^ Since tO VQ = C
sponds the harmonic F_i = -> we have
If ^ he any constant direction whose direction cosines are
cds (\ x) = \ , cos (^ y) = % , cos (^ *) = »!,
£^+*$if*l?
and J- ^ is a harmonic of degree - 2, and to it corresponds the
harmonic, «
93) ^ ®;
which is of the first degree. Since ^2+ mf + V = l, the harmonic
contains ftw arbitrary constants, and multiplying by a third, 4, we
have the general harmonic of degree 1, in the form
94) "i
If in like manner ^, hs, . . . hn, denote vectors with direction
cosines Z2, m2, W2, . . • lnf mn> nn-
d
is a spherical harmonic of degree - (n + 1) and to it corresponds
95) ^ = ^+1ii-i(7>
a harmonic of degree n, and since every h introduces two arbitrary
constants, multiplying by another, A, gives us 2n + 1, and we have
the general harmonic of degree n in the form,
The directions ^, fc2, . . .'*„ are called the ewes of the harmonic.
To illustrate the method of deriving the harmonics we shall find the
first two.
143, 144] FORMATION OF HARMONICS. 397
A 5 d d
A]~ ^ = [>2 + if
1) Laplace, "Theorie des attractions des sphero'ides et de la figure des
planetes." Mem. de I'Acad. de Paris. Annee 1782 (publ. 1785).
145, 146] ZONAL HARMONICS. 399
Considering this as a function of z let us develop by Taylor's
Theorem,
103) ^=^_/)=
, . f A 1 1 0V /1\
and since tor r = 0, -r = — > — = — l—\>
> d r
«*> J-7 + <-0
Now multiplying and dividing each term by rn+l, we find
1 1
105) ^ = i
where
106) P0=l, P
This is the determination of the constant A, adopted by Legendre,
for the reason that, since by the binomial theorem, for r' r we find
^Z 0*' I i*' , 7*' ^
In order to find Pn as a polynomial in [i we may write ^ as
and develop by the binomial theorem.
109)
5 = 0
Developing the last factor,
400
VIII. NEWTONIAN POTENTIAL FUNCTION.
Picking out all the terms for which s -\- 1 = n we get for the
coefficient of f— J
p __
"
2.4(2n-l)(2w-3)
The first polynomials have the values
-30^+3),
147. Development in Spherical Harmonics. We may use
the formula 6), § 128, for an internal point, to obtain the development
of a function of 9, Ym, |I = _ mr»- ir,,,
we obtain, since
118) FM(P) - ±
00
If the coordinates of P he rf, -9-', cp', we have,
while on the right we have an infinite series in powers of r', with
definite integrals as coefficients. Since the equality must hold for
all values of rr less than r, we must have, collecting in terms in r's
119) o o
*m(&> (P') =
o o"
that we have for the values of the integral
120) / IYm(&,
) = r0 + r1 + y2+....
Multiply both sides by Pn(p) sinftdftdcp, and integrate over the
surface of the sphere and since every term vanishes except the nih
we obtain
Tt iTt
123) fff(», V) P.GO sin 9 d» dy = ^ Yn (»', «p'
0 0
124) Yn (»', J) = ~f(^, 9) P« W sin * d» d
,
0 0
actually represents the function f(&, qp'). This theorem was demon-
strated by Laplace, but without sufficient rigor, afterwards by Poisson,
and finally in a rigorous manner by Dirichlet. A proof due to Darboux
is given by Jordan, Traite d' Analyse, Tom. II, p. 249 (2me ed.).
148. Development of the Potential in Spherical Har-
monics. In investigating the action of an attracting body at a
distant point, and for many other purposes connected with geodesy
and astronomy, it is convenient to develop the potential function in
a series of spherical harmonics. \$ x,y,z denote the coordinates of
the attracted point P, r its distance from the origin, a, b, c the co-
ordinates of the attracting point Q, r1 its distance from the origin,
d the distance between them, dr' the element of volume at Q, we have
and using the value of -=- from 105), when r > r',
.< +
which, on removing the powers of r from under the integral signs,
is the required development in spherical harmonics,
126) F=£ + 5 + 7* + "-
where the surface harmonics Yn are the volume integrals
127) T
taken over the space occupied by the attracting body. Since /i enters
into the integrand, and, according to 121), it contains the angular
coordinates #, qp of P, the surface harmonics Yn are functions of &
and (p.
147, 148] DEVELOPMENT OF POTENTIAL. 403
If the body is homogeneous, and is symmetrical about an axis
of revolution, since V is independent of #, y, it is evident that all
the harmonics are zonal, and we have
128) F=^^ + ^^ + ^i + .
r r2 rs
where every Pn is the zonal harmonic in cos #.
If we know the value of V for every point on the axis of
revolution, so that we can develop it in powers of — as
129) F,=0 = ^(r)==|* + i + £*.+... '
then putting cos#=l in 128) and comparing with 129), we find
•4* = Bn so that V is completely determined as
130) 7= + + + .... ; ~:
If in addition the body has an equatorial plane of symmetry, so
that F(cos#) = F(— cosfl-), evidently the development will contain
only harmonics of even order. As a case of this we shall develop
the potential of a homogeneous ellipsoid of revolution in § 161.
Whether the body is homogeneous or not, we may easily obtain
the physical significance of the first few terms in 126). For making
use of the values in 111) since n = ax++c* we have
131) r'Pii — (a
There occur in the first three terms the volume integrals
J J J ^dad^dc==My I J I Qadadbdc = Ma,
jjJQbdadbdc = Mb, jffycdadbdc = Me,
///<
A + B-C
gabdadbdc = F,
26*
404 VIII. NEWTONIAN POTENTIAL FUNCTION.
where M is the mass, a, l>, c the coordinates of the center of mass,
A, B, C, D, E, Fj the moments and products of inertia of the body
at the origin. If we choose for origin the center of mass, and for
axes the principal axes of inertia at that point, we have
a = b = c = D = E=F= 0,
so that the second term of the development disappears, and the third
simplifies, so that we have
* -* -2C)z*
In all these developments, it is to be borne in mind that r is
greater than the greatest value of rr for any point Q in the body.
If the body is a homogeneous sphere, all terms disappear except
the first. If the attracted point is at a considerable distance compared
with the dimensions of the attracting body, or if the body differs
but slightly from a sphere, the terms decrease very rapidly in
magnitude, so that the first is by far the most important. Thus
under these circumstances bodies attract as if they were concentrated
at their centers of mass, or were centrobaric (§ 125). The correction
is in any case in which we are dealing with the actions of the planets,
given with sufficient accuracy by the second term in 133), from
which the moments causing precession were calculated in § 96. In
§ 161 we shall see how the terms depend upon the ellipticity of an
ellipsoid of revolution.
149. Applications to Geodesy. Clairaut's Theorem.
Although, as has been stated, the development 125) is not in general
convergent inside of a sphere with center at the origin which just
encloses the attracting body, on account of the divergence of the
series 105) when rr > r, still it may occur that the performance of
the integrations in 125) causes the latter series to converge even
within this sphere. At any rate for a body having the properties of
the earth, it has been shown by Clairaut1), Stokes2), and Helmert3),
that the series 125) converges at all points on the surface of the
body, and also that for the earth the two terms in 133) represent
the attraction with quite sufficient approximation for applications to
the figure of the earth. In order to exhibit the surface harmonics
1) Clairaut, Theorie de la Figure de la Terre, tiree des Principes de
I'Hydrostatique. Paris, 1743.
2) Stokes, "On the Variation of Gravity at the Surface of the Earth.'
Trans. Cambridge Phil. Soc., Vol. VHI, 1849.
3) Helmert, Geoddsie. 1884.
148, 149] CLAIRAUT'S THEOREM. 405
in terms of angular coordinates, let us introduce the geocentric
latitude ty = — — Q- and longitude qp, in terms of which
x = r cos i[> cos qp, y = r cos ^ sin cos2 = 1 — sin
reduces to
134) C-(l-
1-1- cos 2 op .9 1 — cos2op
2 - — 2
In order to deal with the apparent gravity g, we have to add
to Vj the potential of the attraction that of the centrifugal force, as
in § 123, 73), putting
135) y Vc = i co2 (>2 + y*) = { G32r2 sin2 #.
It is to he noticed that by writing
136)
Fc is itself exhibited as — o2r2 plus a spherical harmonic.
If we now write
^ 2
K= -M~
we have the approximate expression for the potential of terrestrial
gravity
137) U=
with
If the surface of the earth is an ellipsoid whose radius vector
differs at every point from that of a sphere by a small quantity of
the first order, the angle between the normal and the radius vector
406 VIII. NEWTONIAN POTENTIAL FUNCTION.
will be small, its cosine will differ from unity by a small quantity
of the second order, neglecting which we may put
-i on\
139) # = -y^ =
. 9 B- A
,
COS COS
Determinations with the pendulum show that g varies very slightly
with the longitude, we may therefore put B=A, so that
140) U-
1A1\
On a level surface, such as the surface of the ocean will be shown
in § 179 to be,
U= const. = U0.
^M.
For such a surface, equation 140) gives, putting a = ^r and in the
^0
parenthesis substituting a for r, we obtain the equation of the surface,
142) r = al + (l
The substitution of o, for r is permitted in the higher powers because
of the assumption that — differs from unity only by a small quantity 77,
where square is neglected, and thus rm = am(l -\-mrf). Inserting the
value of r from 142) in 141), and approximating in like manner,
143) .9 =
or
4A\ r^
144) g=sr-r n
145) g = g0 (1 + n sin2 ^),
AZ\ 2co2a8 ZK
146) ^ = — ^ — .r-^-
ylf 2a2
The equation 142) is easily seen to be that of an ellipsoid of revo
lution, and putting ^ = 0, ^ = — its semi -axes are found to be
M7\
147)
149] CLAIRAUT'S THEOREM. 407
Accordingly the ellipticity, or flattening (aplatissement, Abplattung),
denned as the ratio of the difference of axes to the greater, is
re~rp ZK
148) e= = «
The quantity c = °^ is equal to the ratio of the centrifugal accel-
eration G) 2 a at the equator to the acceleration of gravity ^—^ at the
same place, while n is equal to the ratio of the excess of polar over
equatorial gravity to the latter. Thus equation 148) gives us Clairaut's
celebrated theorem,
149) e + n = ~c.
, Polar gravity — equatorial gravity
Elhpt^c^ty of Sea -level -\ --
Equatorial gravity
5 Centrifugal acceleration at equator
2 Gravity at equator
The values of the constants in 145) adopted by Helmert as best
representing the large number of pendulum observations that had
been made up to 1884 are given by
150) g = 978.00 (1 + 0.005310 sin2 9),
agreeing closely with the formula given on p. 33. The value of the
centrifugal acceleration is known from the length of the sidereal day,
the time of the earth's rotation, giving
86,164.09 sec.
and the earth's equatorial radius, given by Bessel as 6,377,397 meters.
From this is found
c = 0.0034672 = ^^
giving by 149)
e = ~ x 0.0034672 - 0.005310 = 0.0033580 = — ^-
& u - '
ox dq ox
(W_dVd*qdqj)_ (dV\
dx* ~ dq dx* + dxdx(dq)
In like manner
g2F _
" ~" ~ *
dq dz* "+" \dz) dq*'
dV
_ dV
Accordingly
Q\ ^2
3) T# = -
dV dq\™*dq)
dq
Now since F is a function of q only, the expression on the
right must be a function of q only, say
^gt = h*h* d \ ^ V
"
153. Application to Elliptic Coordinates. Applying this
to elliptic coordinates gives
412 VIII. NEWTONIAN POTENTIAL FUNCTION.
Al _ d L 1 -I/ (gt+l^
"Sir^i K Ql)i^-f)(«
1 I 1 1 1 \ _ m
2 a'-V 2- 2 ~
which is independent of /i and i>, and therefore the system of
ellipsoids A can represent a family of equipotential surfaces. We have
9) w «ji - + +
10) V=AC-
J V(«°
J5 must be such a constant that when A = oo, which gives the infinite
sphere, F= 0. This is obtained by taking the definite integral
between I and oo.
' 00
11) V-A C- =ds
A being taken for the lower limit, so that A may be positive, making
V decrease as A increases. V is an elliptic integral in terms of A,
or A is an elliptic function of F. For
J dl
ia\ A*(d
A \d
a differential equation which is satisfied by an elliptic function.
We may determine the constant A by the property that
lim(VF) = M,
or that ~v
lim (r2 -*rA = — M cos (rx).
r= > i , . ^
We have
^F ^ra^
[by § 73, 86)]
153, 154J ELLIPSOIDAL EQUIPOTENTIALS. 413
From the geometrical definition of A,
lim A = 1.
Now consider, for simplicity, a point on the X-axis, where
$1 = x = r. The denominator becomes infinite in A2, that is, r5, and
so does the numerator. Hence
so that
154. Chasles's Theorem. We have now found the potential
due to a mass M. of such nature that its equipotential surfaces are
confocal ellipsoids, but it remains to determine the nature of the
mass. This may be varied in an infinite number of ways; we will
attempt to find an equipotential surface layer. By Green's theorem,
§ 129, 11), this will have the same mass as that of a body within
it which would have the same potential outside.
If we find the required layer on an equipotential surface S, since
the potential is constant on S, it must be constant at all points
within, or the layer does not affect internal bodies.
The surface density must be given by 10), § 129,
(3 = — — «--> where m is the outward normal to A,
and
3V _ dV dl __, dV
d^i ~ 111 fh^ ~ l ~dl'
Now since
15) (? = - — di —
%TC d'k
Since V is a function of A alone, the same is true of -=T-> which for
a constant value of A is constant. Hence tf varies on the ellipsoid S
as d%. Therefore if we distribute on the given ellipsoid S a surface
layer with surface density proportional at every point to the perpen-
dicular from the origin on the tangent plane at the point, this layer
is equipotential, "and all its equipotential surfaces are ellipsoids confocal
with it. Consequently if we distribute on any one of a set of
confocal ellipsoids a layer of given mass whose surface density is
proportional to d the attraction of such various layers at given
414
VIII. NEWTONIAN POTENTIAL FUNCTION.
external points is the same, or if the masses differ, is proportional
simply to the masses of the layers. For it depends only on A, which
depends only on the position of the point where we calculate the
potential.
Since by the definition of a homceoid, the normal thickness of
an infinitely thin homceoid is proportional at any point to the
perpendicular on the tangent plane, we may replace the words surface
layer, etc., above by the words homogeneous infinitely thin homceoid.
The theorem was given in this form by Chasles.1)
155. Maclaurin's Theorem. Consider two confocal ellipsoids, 1,
Fig. 142, with semi -axes e^, @lt ylf and 2, with semi -axes «2, /32, y2.
The condition of confocality is
16)
Fig. 142.
If we now construct two ellipsoids 3
and 4 similar respectively to 1 and 2,
and whose axes are in the same
ratio & to those of 1 and 3, these
two ellipsoids 3 and 4 are confocal
(with each other, though not with 1
and 2). For the semi -axes of 3 are
wcc-i* v'Lj-i, vV-\ « and 01 4 are /ucc^» i/p^. v^o.
and hence the condition of confocality,
17)
is satisfied. Now if on 3 we distribute one infinitely thin homceoidal
layer between 3 and another ellipsoid for which # is increased by d&,
and on 4 a homoeoidal layer given by the same values of & and d&,
and furthermore choose the densities such that these two homoeoidal
layers have the same mass, then (since these homceoids are confocal)
their attractions at external points will be identical.
Now
the volume of an ellipsoid with axes a, 6, c, is —stabc,
o
that of the inner ellipsoid of the shell 3 is accordingly
and that of the shell is the increment of this on increasing # by
(vol. 3) =
(vol. 4) =
or
Similarly
1) Chasles, "Nouvelle solution du probleme de 1'attraction d'un ellipsoide
h^terogene sur un point exterieur. Journal de Liouville, t. V. 1840.
154, 155, 156] ATTRACTION OF CONFOCAL ELLIPSOIDS. 415
Consequently, if we suppose the ellipsoids 1 and 2 filled with matter
of uniform density ^ and p2, the condition of equal masses of the
thin layers 3 and 4,
is simply
18)
that is, equality of masses of the two ellipsoids. And since for any
two corresponding homoeoids such as 3 and 4 (#• and # -f dti) the
attraction on outside points is the same, the attraction of the entire
ellipsoids on external points is the same.
This is Maclaurin's celebrated theorem: Confocal homogeneous
solid ellipsoids of equal masses attract external points identically, or
the attractions of confocal homogeneous ellipsoids at external points
are proportional to their masses.1)
156. Potential of Ellipsoid. The potential due to any
homoeoidal layer of semi -axes a, /3, y is found to be from our preced-
ing expression for F, 14),
14)
M r
* V vw
where A is the greatest root of
^^
Now if the semi -axes of the solid ellipsoid are a, &, c, those of
the shell a = &a, /3 = #&, y = frc, we have M = k-st^d&abc, if the
density is unity, and
20) d,V **9***dbcJ-
where A is defined by
21)
To get the potential of the whole ellipsoid, we must integrate
for all the shells, and
22) V^Zxalfi fad ft C ds
J J V(aW +
1) Maclaurin, A Treatise on Fluxions. 1742.
416 VIII. NEWTONIAN POTENTIAL FUNCTION.
For every value of # there is one value of A, given by the
cubic 21).
Let us now change the variable 5 to t, where, # being constant,
s = &2t, ds = &2dt, and put A = &2u.
Then
1 oo
/"* /* dt
23) V= 2xabc I &d& I — g 9 >
€x €X
0 w
where u is defined by
24) ^ -
Since &* is thus given as a uniform function of u, we will now
change the variable from # to u.
Differentiating 24) by #,
25)
When ^ = 0, w = oo, and when ^ = 1, u has a value which we
will call 6j defined by
X2 ?/2 22
^rG + ^f^ + ^qr^ =
Accordingly, changing the variable,
27) F=^a&c / \^^ + ^^ + ^^^du C-^=^=^^
The three double integrals above are of the form
28)
where
This may be integrated by parts.
Call
156] POTENTIAL OF ELLIPSOID. 417
Now
W= ff(t}dt,
?>'(«) = - /"(«)•
Inserting' these values in 29),
or the variable of integration being indifferent, we may put u for t
in the first integral.
Applying this to our integral 27), by putting C successively
equal to a2, IP, c2, multiplying by #2, i/2, #2, and adding,
31) V=
Now the first three terms of the integrand are, by definition,
equal to 1, so that
32) V
This form was given by Dirichlet.1)
If the point x, y, s lies on the surface of the ellipsoid,
.
then 0 = 0 and
33) V=itabcf {l_-j£-_--j£ rr-l-: ~^~ =•
^y I az+% o*-}-u c -fu} y^^.uj^^.uj^^_u^
o
We find for the derivatives of F,
oo
fly /* du
^ = CXJ (a2 + W)V(^R)(&2 + ^)(c2-H*)
a
-, dai* x* y* z* \ 1
— xabc^l 1 — asj—Q— pV^ — ^2jr^| /— g 2 8 — — •
1) Dirichlet, "Uber eine neue Methode zur Bestimmung vielfacher Integrate."
Abh. der Berliner Akad., 1839. Translated in Journ. de Liouville, t. iv., 1839.
WEBSTER, Dynamics. 27
418 VIII. NEWTONIAN POTENTIAL FUNCTION.
By definition of (3, the parenthesis in the last term vanishes, and
00
= — 2nabcx
C du
I -
J (a8 + w)l/(a8 + w)(&8 + w)(c8-|-«
a
34) ~ = -2xabcy f —
V9 J (62 + ^)V(a2 + ^)(i
a
co
0 ,
= — ZTtaocs
157. Internal Point. In the case of an internal point, we
pass through it an ellipsoid similar to the given ellipsoid, then by
Newton's theorem it is unattracted hy the homoeoidal shell without,
and we may use the above formulae for the attraction, putting for
a, &, c, the values for the ellipsoid through #, yy 0, say &a, #&, &c.
Since the point is on the surface of this, 6 = 0.
35)
o
Now let us insert a variable u' proportional to u, u =
co
dV
36") -TT— = — 2rt& abcx
v$
ft,
0
The # divides out, and writing u for the variable of integration
00
dV
C du
I - .
J (a2 + ^)]/(^ + ^)(62 + ^)(c2 + ^)
0
So that for any internal point, we put = 0 in the general
formula. Integrating with respect to x, y, 0, we have
00
A *• «•
37) V=xabc \l- -^L--^-^
ft/ »
0
The constant term must be taken as above in order that at the
surface V may be continuous.
In the case of an internal point the above four integrals may
be made to depend upon the first. Calling
156, 157, 158] POTENTIAL WITHIN ELLIPSOID. 419
38) • ' du
o
and accordingly
39) V =
The integral 0 is an elliptic integral independent of x, y, z, and
so are its derivatives with respect to a2, &2, c2. Calling these respec-
,. , L M N ,
tively —9 —) — t we have
^444
40) V=itabc
a symmetrical function of the second order, and since L, M, N are
of the same sign, the equipotential surfaces are ellipsoids, similar to
each other. Their relation to the given ellipsoid is however trans-
cendental, their semi- axes being
V*g^ *
c<&
o
^(tt'2)
We have for the force,
3V
V T
42) yLx,
Therefore, since for two points on the same radius vector,
-17- - T7- - ^ -
X, F, Zt rt
The forces are parallel and proportional to the distance from the
center, though not directed toward the center.
158. Verification by Differentiation. For an outside point
we have, differentiating 34)
a
27'
420 VIII. NEWTONIAN POTENTIAL FUNCTION.
Now by § 73, 86),
06 2x /I x* y* z* \
dx a* + 6/ j(a2+<>)2 ~*~ (62+<02 (c2-f<02j
Forming •«— , and -^ and adding,
QO
C\ 1 1 1 I
a
The integration may be at once effected.
Since
7/ s (du , dv , div]
d(UVW)=UVW\- ->;
, 1 u v w I
we nave
45) d\ , l = =}
(V(a* + u)(b*-\-u)(c* + u)l _
_ _ du _ f ya*+u }
~ y(a* 4- u] (&2 + u} (c2 -f u) I 2 |/(a2 -f u)s I
M 1 i 1 I 1 1 du
2 |a2+tt 62 + ^ c2 + wjy^2^w)(&2_^w)
The integral becomes then
which cancels the second term, and z/F=0.
For an internal point
/*
I -
J (
At infinity tf = oo, and V and its derivatives accordingly vanish.
Therefore the value of V found satisfies all the conditions.
159. Ivory's Theorem. If x, y, z is a point on the ellipsoid (1)
4T) $ + £ + # = 1,
the point
(%i Oj Cj
lies on the ellipsoid (2)
158, 159, 160] IVORY'S THEOREM. 421
These will be called corresponding points. We shall now assume
that these two ellipsoids are confocal, and (2) the smaller. Then
49) V = a22 + A, V = V + *, V = <*' + *•
The action of (2) on the external point x, y, z is
50) X2 = - 2
v + v +
we must have (5 = 'k.
If now we substitute
51) X2 = — 2y7taJ)<,Cc>x I - u
J («i 2 + «') V(oi 2 + «')
o
Now the attraction of the ellipsoid (1) on the interior point
«« 5« c2 .
> y^> 2 is
ax f-.^ q
cc
52) Zj = - 277ca1b1cLx^ C - du
so that
57) r~,aVf- -W-W --
^7 ss(62- a!)V(6s- a8) (1 - s8)
ds
no\ TT-
=
59) X = 2*aVxT ___
J S8(ft2
4:7cab2x / s*ds
~A / " ~
(&2-a2)2,y (l-s2)
Now
/s2ds s
^l==W
160]
so that
60)
61)
ELLIPSOIDS OF REVOLUTION.
X =
s8(62-a2)2
'/
4«a68y / s2^s
= (»._a,v '|7
No
w
s*ds
so that
62) Y=
for if
then
-,/&«_ a* -, /62-a
Jbor sm~~] I/ 2 we may write tan~] I/ 2
423
These formulae all serve for an oblate spheroid, where a < 6.
For a prolate spheroid, "b > a, they introduce imaginaries, from which
they may be cleared as follows.
Call
sin— 1 (iu} = #,
then
iu = sin &, 1/1 -}- u2 = cos -91,
therefore
Put
cos # — i sin # = ]l + u2 -f w,
sin—1 (iu) = & = i log {
u.
424 VIII. NEWTONIAN POTENTIAL FUNCTION.
Therefore
64)
65) y =
(a'-&2)2
..!
In all these formulae, 6 is the larger root of the quadratic
66) _-L + _£_!,
for an outside point, and 6 = 0 for an inside point. In the latter
case, we have functions only of the ratio -v--1)
161. Development of Potential of Ellipsoid of Revolution.
We may develop the expression 58) for the potential of an oblate
ellipsoid of revolution in a series of spherical harmonics. Considering
first a point on the axis of revolution, let us put
x = r > a, y = 0,
so that we have by 66), 61),
67) -^- = 1, J = 0
«2-|-(7
and using the tan"1 instead of sin"1 in 58) and 60), we obtain
,
2 tan~]
2
~Qx Jr
68) Vk =
Remarking that M the mass of the ellipsoid is -nab2, and
developing the antitangent, we have, if r > a > ]/62 — a2, 62 <
3M
= 3 M I ^, (- 1)" (fe2 - ay ^, (- 1)* (&2 - a2)"
2 }^/2w + l r2w+A ^-/2n + 3
1) Thomson and Tait, Natural Philosophy, Part H, § 527.
160, 161, 162] ENERGY OF DISTRIBUTIONS. 425
from which, by § 148, 130), -we obtain finally,
70) F.g
and this series is convergent for points on the surface of the ellipsoid
itself, if &2 < 2 a2. The series converges extremely rapidly if ^ differs
little from unity.
162. Energy of Distributions. Gauss's theorem. If a
particle of unit mass be at P, (x, y, g) at a distance r from a particle
of mass m(J, the work necessary to bring the unit particle from an
infinite distance against the repulsion of the particle mq will be
71) W=7mJ=yV(X,y,z)=rrp.
If, instead of a particle of unit mass, we have one of mass mp
the work necessary will be mp times as great,
m
72) Wpq = r-?mp = ympVp = ymqVq,
where _ %
**-%
In other words, this is the amount of loss of the potential energy
of the system on being allowed to disperse to an infinite distance from
a distance apart r. Similarly, for any two systems of particles mp, mq,
73) wft =
Vp being the potential at any point p due to all the particles q
and Vq being the potential at any point q due to all the particles ^?.
This sum is called the mutual potential energy of the systems p and q.
If however we consider all the particles to belong to one system,
we must write
where every particle appears both as p and q, the — being put in
because every pair would thus appear twice. This expression has
been given in § 28, 36).
If the systems are continuously distributed over volumes t, t1
we have
75) Wpq =
426 VIII. NEWTONIAN POTENTIAL FUNCTION.
The theorem expressed by the equality of the two integrals is
known as Gauss's theorem on mutual energy, where Vp' represents
the potential at p due to the whole mass Mq, Vqj that at q due to
the whole mass Mp})
The above equality may be also proved as follows. Since
76) Qf--{
and
the triple integrals in 75) become respectively,
77) ~-4
and
, /»/»/»
Vq^Vq'dlq.
Now since outside of r, 4V = 0 and outside of T', z/Ff = 0 the
integrals may be extended to all space. But by Green's theorem,
both these integrals are equal to
dV dV dV dV 3V cV
dx dx dy dy dz dz
since the surface integrals
on
vanish at infinity. Gauss's theorem accordingly follows from Green's
theorem and Poisson's equation.
163. Energy in terms of Field. For the energy of any
distribution consisting of both volume and surface distributions, the
sum 74) becomes the integrals
78) W =
Now at a surface distribution Poisson's equation is
1 f dV cV\
•"1
1) Gauss, uAllgemeine Lehrsatze in Beziehung auf die im verkehrten Ver-
haltnisse der Entferming wirkenden Anziehungs- und Abstossungskrafte." Werke,
T» J TT _ + r\m
Bd. Y, p. 197.
162, 163, 164] ENERGY OF FIELD. 427
If, as in § 131, we draw surfaces close to the surface distribu-
tions, and exclude the space between them, we may, as above, extend
the integrals to all other space, so that
the normals being from the surfaces S toward the space r. But by
Green's theorem, as before, this is equal to the integral
»> "- £///!«©'+ ©'+©><•
00
Thus the energy is expressed in terms of the strength of the field
at all points in space. This integral is of fundamental importance
in the modern theory of electricity and magnetism.
It is at once seen that this always has the sign of y, so that it is
positive for electrical or magnetic, negative for gravitational dis-
tributions.
CHAPTER IX.
DYNAMICS OF DEFORMABLE BODIES.
164. Kinematics. Homogeneous Strain. We have now
to consider the kinematics of a body that is not rigid, that is, one
whose various points are capable of displacements relatively to each
other. In the general displacement of such a body every point x, y, z
moves to a new position x\ y', z1 , so that xf, y', 0' are uniform func-
tions of x, y, 0. The functions must also be continuous, that is, two
points infinitely near together remain infinitely near together, unless
ruptures occur in the body.
The assemblage of relative displacements of all the points is
called a strain. The simplest sort of strain is given when the func-
tions are linear, that is,
x1 = a^x -f a2y -j- a^z,
1) y' = \x + \y + M,
0' = W + %y -f c3z,
where the a's, Vs and c's are nine constants.
428
IX. DYNAMICS OF DEFORMABLE BODIES.
No constant terms are included because a displacement represented
by x' = a, y' --=~b, #' = c, would denote a translation of the body as
if rigid, which is unaccompanied by relative displacement or strain.
Let the solutions of the equations 1), which we shall term the
direct substitution, be
2)
where
ABx'
. . ,
— z/ etc.
— z/', etc.
4ii ft*
I .' C2 j CS
A strain represented by the equations 1) is said to be homogeneous.
If the accented letters denote initial positions, and the unaccented
letters final positions, the strain represented by equations 2) is said
to be inverse to the first strain.
In virtue of equations 1) or 2) a linear relation between x, y, z
becomes a linear relation between x1, y', z\ Accordingly in a homo-
geneous strain a plane remains a plane, and a straight line, being
the intersection of two planes, remains a straight line. Finite points
remain finite, since the coefficients are finite, accordingly parallel
lines, intersecting at infinity remain parallel. Parallelograms remain
parallelograms (their angles being in general changed), and therefore
the changes of length experienced by equal parallel lines are equal,
and for unequal parallel lines proportional to their lengths. Thus
any portion of the body experiences the same change of size and
shape as any equal and similarly placed portion at any other part
of the body. This is the meaning of the term homogeneous, which
signifies alike all over.
When two vectors OP of length r and OP' of length r' drawn
from the same origin are so related that their respective components
x, y, 2, x\ y\ z' are connected by the equations 1) or their equi-
valents 2) either vector is said to be a linear vector function of the
other. The properties of such linear functions are of great importance
in mathematical physics, and will now be taken up before their
application to strain.
164] HOMOGENEOUS STRAIN. 429
Let us examine the conditions that the two vectors OP and OP'
shall have the same direction. The condition for this is
where A is to be determined. Introducing the values x' = &x, y' = ky,
%] = fig into equations 1) we obtain
(% — A) x 4- a2y 4- a3 # = 0,
4) \x + &-A)y 4-M = 0,
c^ + c2# 4- (cs — A) £ = 0,
a set of linear equations to determine x, y, z. The condition that
these shall be compatible is that the determinant of the coefficients
vanishes.
5)
, 6,
0.
This is a cubic in A. Let its roots be Alt A2, A3. Inserting any
one of these in 4) we may find the ratios of x, y, z giving the
direction of the vectors in question.
Supposing that A1; A2; A3 are real, let us find the condition that
the three directions are mutually perpendicular.
Substituting first I = ^ and then I = A2 in 1) and 3), we have,
denoting the values of x, y, z by corresponding subscripts,
+ Vi^
i + &2«/i -+ ^^ =
^i +
-f
Multiplying the first three respectively by #2,2/2,'#2 and adding, and
subtracting the sum of the last three multiplied respectively by
x # we
7) fe - &8) (2/1^2 - *i 2/2) 4- (OB ~ ci) (^i ^2 - %^2) 4- (&i- «a
= (A! - A2) fe^ + 2/^2 4- ^^2).
The condition for perpendicularity of rlf r2 is
% 4- /«/ + ^^ = 0.
430
IX. DYNAMICS OF DEFORMABLE BODIES.
Accordingly the left-hand member of 7) must vanish. If rs is
perpendicular to t\r2 its coordinates must be proportional to their
vector -product. Thus we may write 7) as
Inasmuch as the order of suffixes 1, 2, 3 is indifferent, if the
three vectors r19 r2, rs are to be mutually perpendicular, equation 8)
must be satisfied by the components of all three. This can be true
only if we have
that is, the determinants of the substitutions 1) and 2) are symmetrical.
In this case the linear vector -function is said to be self - conjugate,
and a strain represented by such a function is called a pure strain.
165. Self -conjugate Functions. Pure Strain. We will
consider this important case in detail. Adopting a symmetrical
notation, let us write . 7
0 = ax + hy + gs,
9) y' = hx + l)y -f fz9
z1 = gx 4- fy+ cz.
If by cp we denote the homogeneous quadratic function
10) fp = axz -\- by* -f £#2 + %fy& + 2g2x + 2hxy,
equations 9) may be written
so that the vector OP' (Fig. 143) is parallel
to the normal at the point P, whose co-
ordinates are x,y,z, lying on the quadric
cp = + E2} where E is a constant introduced
merely for the sake of homogeneity. In
like manner calling
Fig. 143.
11) (p' =
-f
equations 2) are
-f 2
X
12)
so that OP is parallel to the normal at P', whose coordinates are
x', y', z', a point on the quadric 166] PURE STRAIN.
431
By means of either of the quadrics 9 and
, if/ coincide
in direction.
Multiplying together 16) and 22) we obtain
24) <.9'=S2,
and, since the directions of (> and p' coincide for the axes of either
ellipsoid , we see that the ellipsoids are reciprocal with respect to a
sphere of radius S. Multiplying 15) by 16) and 22) respectively we
get for the axial directions ,
165]
STRAIN- QUADRICS.
433
that is, the axes of the ellipsoids i(j, ij> are proportional to the squares
of the axes of the quadrics tp and g>'. By means of either pair of
quadrics
* _
~
1 Off X
m^^ii
1 dcp' y'
2 "dy b
1 d(p' z'
¥ U7 = ~c
In the cubic for the axes of
The determinant of the coefficients of the substitution 9) accordingly
represents the ratio of expansion, and since parallel lines are stretched
V
in the same ratio, the ratio of expansion of volume -=• is everywhere
the same.
165, 166] SIMPLE ROTATION. 435
166. Rotation. Let us return to the case of the general
homogeneous strain given by equations 1) and let us find the condi-
tion that all points situated before the strain on a sphere with center
at the origin remain on the same sphere after the strain.
The condition
x'* + y'a + e'* = x* + f + £
gives
31) fax -f a.2y + a3#)2 -f (\x + bzy -f &3#)2
which being true for all values of x, y, s necessitates the equality of
the coefficients of corresponding sqares and products on both sides
of the equation , that is,
V -I- If + c,2 = 1,
32) aS + V + cS-*!,
a^a^ -f &i&2 -f
33) V3 + &2&3 +
Equations 32) show that a, &, c with the same suffix are direction-
cosines of a line, equations 33) show that the three lines are mutually
perpendicular, in other words, the equations of strain are merely
those of transformation of coordinates, and the result of the strain
is merely a rotation of the body as if rigid.
Let us obtain the analytical expression for an infinitesimal
rotation about an axis. Let the direction -cosines of the axis be
X, [i, v and the angle of rotation be do. Since we have proved in
§ 57 that infinitesimal rotations may be resolved like vectors and
treated like angular velocities, we have the components^ of rotation,
34) cox = ^do3, G)y = /idea, oz = vd&,
from which by equations 119), § 76, we obtain the infinitesimal
displacements,
x1 — x = dx = z&y — ycoz = (0fi — yv) do,
35) y1 — y = dy = x&z — ZG)X = (xv — gfy do,
#f -- 8 = dz = ox — xc3 = k — %i do.
From this we obtain the substitution for the rotation considered
as a strain,
28
,
x = 1 • x — voo - y +
36) y1 = vdo -x+l-y- Udo
436
IX. DYNAMICS OF DEFORMABLE BODIES.
The determinant of the substitution is skew symmetrical.
The ratio of expansion is
37)
V
~V
Ado ,
Ado , 1
which is equal to 1 plus terms involving do2 which are negligible,
agreeing with the result that rotation as a rigid body is unaccompanied
by change of volume.
167. General Small Strain. We shall now consider small
strains in general, that is, strains in which the displacements of all
points are small quantities whose squares and products may be
neglected.
Let the components of the displacements or shifts be
38)
x — x = u, y — y = v, s — z = w,
so that if we now write the coefficients of the strain as
39) y' = \x 4- (1 -
we have the shifts given by the substitution
a.2y
40)
w=
and if the a's, &'s and c's are small, u, v, w will be small quantities
of the same order. The ratio of dilatation is by 30)
41) =
The quantity
42)
terms of higher order,
1 4- &i 4- &2 + cs + terms of higher order.
IT! -\r
is the increment of volume per unit volume, and will be referred to
simply as the dilatation and denoted by (5.
166, 167]
COMPOSITION OF SMALL STRAINS.
437
Suppose two small strains take place successively according to
the equations 39) for the first, and
X'
43)
for the second.
Substituting the values of x\ y'} #r from 39) in 43) we obtain
*4(l+<
-f
-f
Neglecting terms of the second order we obtain the equations of the
resultant strain
-f- -f
44) ?" = &
and for the resultant shifts
45)
(a2 -f
that is, successive small strains are compounded by adding their
shifts. This important proposition enables us conveniently to resolve
small strains into types already studied. Every small strain represented
by equations 40) can be written by addition and subtraction of
equal terms
u = a^x + (aa -f &x)y + (a3 + cJ0
46)
-f
438 IX- DYNAMICS OF DEFORMABLE BODIES.
Accordingly we may write the strain as the resultant of two,
U = U ! + u-2 9 V = vi + ^2 J W = W-L + W2,
where
denoting a pure strain, and
48) v2 = — (^ — a2) x — — (c2 -
denoting a rotation CD whose components are
49) OIT = -9(03 —
Thus every small strain may be resolved into a pure strain and a
rotation.
In order to bring out the symmetry let us write the pure strain
% = sxx + gzy + gyz,
50) v1 = gzx + syy + gx0,
MI = 9y% + 9xy + 5,-er,
where
Thus the six quantities g and co are respectively the half sums
and half differences of shift -coefficients symmetrical about the main
diagonal.1)
1) In the usual notation the #'s are defined as the above sums without the
coefficient — i as stated by Todhunter and Pearson, A History of Elasticity and
Strength of Materials, Vol. I, p. 882, "The advantage which would arise from
167, 168J TYPICAL SIMPLE STRAINS. 439
The general small strain is accordingly completely defined by
the nine small coefficients,
sx, sy, ss, gxj gy, g,, a*, &y, G>Z.
168. Simple Strains. Stretches and Shears. The pure
strain 50) may be resolved into two parts
(a) (b)
52) vl = syy, <=^-f 0 + gx0,
wj = szs, Wi" = gyX + gxy + 0.
A strain whose equations contain but a single constant is called a
simple strain. Thus we may resolve the strain (a) into three simple
strains of which the first is given by
u = sxx} v = 0, w = 0.
This represents a displacement in which each point is shifted parallel
to the x -axis through a distance proportional to its x coordinate.
Such a displacement is called a stretch. The constant sx represents
the distance moved by a plane at unit distance from the YZ- plane
and measures the magnitude of the stretch or the linear expansion
per unit length. If s is negative the stretch becomes a squeeze.
The strain (a) accordingly represents the resultant of three
simple strains, namely stretches, of different amounts in the directions
of the coordinate axes, which are evidently the axes of the strain.
The semi- axes of the strain -ellipsoid are 1 + sx> 1 + sy, 1 -f sz and
its equation
? "• 7i i 0 N2 === •'J
or neglecting squares of small quantities,
(1 - 2s*) x* + (1 - 2sy) ^ -f (1 - 2s,) £ = 1.
The dilatation is by 42)
53) 6 = sx + sy + sz.
Obviously we can have 6 = 0 if at least one of the stretches is
replaced by a squeeze. If the three s's are equal we have a simple
introducing the — into the slides is thus obvious", and we have therefore so
introduced it, although to them "it seemed too great an interference with the
nearly general custom.'1 We have also introduced a single suffix, gx, instead
of the more usual double suffix notation, g z, feeling that the brevity and
analogy with a>x thus gained justifies the change.
440
IX. DYNAMICS OF DEFORMABLE BODIES.
strain known as a uniform expansion for which the strain -ellipsoid
is a sphere and the dilatation = 3s.
The part (b) of the pure strain represents a strain which, like
a rotation, is unaccompanied by a dilatation, but which differs from
a rotation in that it involves a change of form. We shall consider
it, as we have the part (a) in three parts.
. In the first,
u =
v = gzx, w =
G
every point is shifted in the X direction through a distance pro-
portional to its distance from the XZ-plane, while it is shifted
in the Y direction through a distance proportional to its distance
from the YZ- plane. Points at
unit distance from the two
named planes are shifted both
ways by the same amount gx,
so that the new positions of
the planes XZ, YZ make
with the old angles whose
tangents or sines are equal
to gz.
The square OACS (Fig. 145)
becomes the rhombus OA'C'B',
which is symmetrical about
the diagonal OC bisecting the
angle XOY. The diagonals
AS and OC maintain their
directions unchanged, and are
accordingly two of the axes of the strain, the axis OZ being the
third. The stretch -ratio along OC is
Fig. 145.
OC'-OC CC'
OC
OC
C"C
BC
SB'
OB
as may be seen by inspection of the figure. The stretch along the
perpendicular axis OE is negative,
OE'-OE EE' EE"
OE
OE
AE
The stretch along the ^-axis is zero. Accordingly the sum of the
three stretches along the axes of the strain is zero. Such a strain,
involving a distortion but no expansion and depending upon a constant #,
is called a simple shear. The plane of the shear is the plane parallel
to which all points are displaced, in this case the XY- plane.
A shear may be defined as a stretch along one axis combined
with a squeeze of equal magnitude along a perpendicular axis, and
168, 168 a] SHEARS. 441
zero stretch along the axis perpendicular to both. The shear just
considered is a pure shear, that is, without rotation. It is easily
seen that the above shear might have been obtained if all planes
parallel to XOZ had been moved parallel to themselves a distance
in the X- direction equal to 2gzy, giving the rotational shear
u' = 2g,y, v' = 0, w' = 0
and then rotating about the Z-axis through an angle az=gz, accord-
ing to the equations
The lines OAf, OB', which before the strain were perpendicular,
have respectively the direction- cosines 1, gz, 0 and gz, 1, 0 and the
cosine of their included angle is accordingly 2gz. This change of cosine
which, as we have just seen, is equal to the amount of sliding of
the plane at unit distance from XOZ is commonly called the amount
of shear, so that the stretch and squeeze of the axes are each one
half of the amount of the shear.
We may now define the strain (b) as a combination of three
simple shears of amounts 2gx, 2gy, 2gz, with planes mutually perpen-
dicular and equivalent to stretches of amounts gzj gx and gy along
the bisectors of the angles XOT, YOZ, and ZOX respectively, which
make angles of 60° with each other, together with squeezes of the
same amounts along the bisectors of the other angles. We have
thus the final positions of six points on these lines, or just sufficient
to determine an ellipsoid whose center is given. This is the strain-
ellipsoid.
The strain (b) will be called a general shear. The quadric cp is
54) y = x2 + f + £ + %gxys + 2gvxe + 2g,xy = B2,
and the shears are the coefficients of the product terms. If the
equation of the quadric is transformed to its principal axes the
product terms vanish. Accordingly we may always find three mutually
perpendicular axes with respect to which the shear components vanish.
These are the axes of the shear. (It may be remarked that the
equations of the general rotational shear are obtained from 1) by
putting a± = &2 = CB = 1.)
In order to distinguish between the geometrical term shear and
the dynamical shearing stress, to be presently considered, it will be
convenient to characterize the coefficients g as the slides (corresponding
to the French glissementj German Gleitung).
168 a. Elongation and Compression Quadric. Since the
equations 50) for the shifts, the components of the vector dis-
placement q, as a function of r are of precisely the same form as
442 IX- DYNAMICS OF DEFORMABLE BODIES.
equations 9) for r we have the complete geometrical representation
here applicable. Of the four quadrics the first is the most important.
The length r of the line OP is changed by the strain to r' which,
when the strain is small, differs from it by a small quantity, so that
the stretch
~ r> \ r' — r
05) Sr = -^—
is a small quantity of the first order. But, since the angle between r
and r' is infinitesimal, we have to the first order, if q is the dis-
placement PP',
56) r' = r -f gc
-~ _ gcos(gr) _
r r2
Now if UjVjW are given by equations 50) the numerator becomes
58) r*sr = sxx2 + syy* -f szz* + 2^^ + 2^£# + 2g,xy = #.
If we put this equal to unity we have
59) ^ = ~2>
where r is the radius vector of the quadric
60) z-l.
This is called the elongation and compression quadric, and it is to be
noticed that the displacement of any of its points is in the direction
of the normal, for
/»w\ l 3% l d% 1 d%
61) M = -^, #= *, ^= *
% ox % dy 2 cz
Since any one of the six coefficients may be positive or negative,
the quadric may be an ellipsoid or an hyperboloid. In the latter
case not all the lines drawn from the origin will meet the surface,
and for those which do not r is imaginary and sr is negative.
If we construct the conjugate hyperboloid, % = — 1, those rays
which do not meet the first hyperboloid meet this, and the magnitude
of the compression is given by
62) *<.—.£•
Lines that meet both hyperboloids at infinity and therefore have a
zero stretch or compression lie on the cone % = 0, asymptotic to the
two hyperboloids, and known as the cone of no elongation.
All lines which are equally elongated with the stretch S, where
63) S = i {sxx* + syif + s,s* + 2gxyg + 2gysx + 2gzxy],
lie on the cone
168 a] ELONGATION QUADRIC. 443
64) (sx — £) x* + (s9 — S)y2 + ($y — S)z*+ 2gxyz + %gyxz + 2ggxy = 0,
which may be called a cone of equal elongation S, of which the cone
of no elongation is a particular case.
Let us form the elongation quadrics for expansions and shears.
If the slides vanish we have
65) x = SxX* + s,y2 + s,s* = 1,
and for a simple stretch in the X- direction
66) sxx* = 1,
the elongation quadric breaks up into the two parallel planes,
l^SxX —1=0 and y ' sxx +1=0,
at distances + — from the origin.
Since for any line making the angle # with the X-axis we have
r = — = 9
ysx cos #
the stretch is given by
67) sr = -g = sx cos2 #.
The cone of no elongation is therefore the plane & = ^ parallel to
the above pair of planes. In equation 65) if sx) sy, sz are of the same
sign the quadric is an ellipsoid and the cone of no expansion is
imaginary. If one s has a sign different from that of the others we
have two hyperboloids and the cone of no expansion is real and
separates the stretched from the squeezed lines.
In the general shear sx = Sy = sz = 0 we have
68) x = 2 (gxyz + gyex + g»xy) = 1,
and the cone of no elongation
69) gxyz 4 gysx + gexy = 0,
contains the three coordinate -axes as generators. These are therefore
unstretched. In a simple shear parallel to the XY- plane we have
70) % = 2gzxy = ± 1
which represents equilateral hyperbolic cylinders with axes bisecting
the angles between the x and y axes. The cone of no elongation,
xy = 0, breaks up into two coordinate -planes, x = 0 and y = 0.
These two planes are undistorted, and are the planes of circular
section of the strain -ellipsoid.
444 IX. DYNAMICS OF DEFORMABLE BODIES.
A combination of two simple shears in planes at right angles
obtained from 52 b) by putting gz = 0, has the elongation quadric
71) gxyg -f gyxg = 0,
which breaks up into the two planes
72) # = 0 and gxy
at right angles to each other. It is to be noticed that the cone of
the resultant of three simple shears in mutually perpendicular planes
does not so break up.
We have seen that we require nine constants to specify the
general homogeneous strain, of which three belong to the rotation,
six to the pure strain. Let us consider the number of data required
to specify a simple pure strain. To specify a uniform dilatation we
require only the constant of dilatation tf; for a simple stretch, the
direction of the axis, involving two data, and the magnitude of the
stretch, making three in all; for a simple shear, four data, the
magnitude of the shear, two to fix the plane of the shear and one
additional for an axis. Consequently we may always represent a
general strain as the resultant of three simple expansions, or of two
simple shears and a uniform dilatation.
169. Heterogeneous Strain. If the displacements are not
given by linear functions of the coordinates, the strain is said to be
heterogeneous. In this case we may examine the relative displacements
of two neighboring points. Let the coordinates of the first point P
be before the strain x, y, z, and after it x -f u, y + v, z + w, and
those of the second, §, be before x + f, y + g, 0 + h, and after
% + f -\- u', y 4- g -4- v\ z + h + w1. If the point Q be referred to P
as an origin both before and after the strain, it has as relative co-
ordinates before f, g, h, and after f -f u' — u, g + vf — v, h -f w' — w,
so that the relative displacements are u' — u, v' — v, wf — w. Now
u, v, w may be any functions of the coordinates x, y, 3 of P, but
they must be continuous, otherwise the body would be split at
surfaces of discontinuity. Accordingly u\ v',w* being the values of
u, v, w for x -\- ff y -\- g, % -\- h may be developed by Taylor's theorem,
so that, neglecting terms of order higher than the first in /*, #, li
i ~c-u . du . ^ du
u — u = f-x — \- g- — h » -o->
1 dx ' y oy ' dz
rroN f r ^V , fiv . •* dv
73) v ' — v = fjr- + g^r -f h -5-9
1 dx ' y dy ' dz
3w , 7 dw
168 a, 169] HETEROGENEOUS STRAIN. 445
Thus the relative displacements are given as linear functions of the
relative coordinates f, g, li whose coefficients are the values of the
nine first derivatives at the point P, that is to say, constants for all
points Q in the neighborhood of P, consequently the relative strain
of the portion of the body in the neighborhood of P is homogeneous.
Thus we say that any continuous heterogeneous strain is homogeneous
in its smallest parts.
Comparing with equations 49) and 51) we find the stretches,
dilatation, slides and rotations at any point to be respectively
„ A\ du dv dw
74) sx = -, Sy = -, S, = -,
75)
'
ox
1 dw . dv\ 1 du dw\ l dv . du
1 (w . v\ 1 (u , w\ l (dv . du
?.-¥W + J^ ^ = ¥\^ + W 9*=*(dx+Jy
1 /dw dv\ 1 (du dw\ l (dv du
X== ~ \-~ -- -x— ?
2 V^i/ W
= — ^ -- -o— > G) * == — ^ -- -K—
2 \^^ &ar/ 2
Thus the volume dilatation is equal to the divergence of the dis-
placement, while the rotation is equal to one half its curl.
We might have obtained the value of 0 by the divergence
theorem. Consider any closed surface S fixed in space so that por-
tions of the deformable body flow through it daring the strain, and
let us find the volume of the matter which passes outward through S.
Through an element dS at which the displacement is q there passes
out a quantity filling a prism of slant- height q and base dS whose
volume is therefore qcos(nq)dS, where n is the outward normal
to S. Through the whole surface there accordingly issues an amount
whose volume is
78) Q = I I qcos(nq)dS
= I I [u cos (nx) + v cos (ny) -f w cos (n&)} dS
by the divergence theorem. This is accordingly the increase in volume
of the portion of substance originally included by the surface S.
The ratio of this to the original volume is accordingly the mean
value of the divergence in the volume in question, and making the
volume infinitesimal, this becomes the dilatation 6.
In order that a strain shall be everywhere irrotational we must
have the curl components of the displacement vanish everywhere.
446 IX- DYNAMICS OF DEFORMABLE BODIES.
But by § 31 this is the condition that the displacement is a lamellar
vector and
' dxy dy' ~ dz
Then (p is called the strain -potential. Only when the strain is ir-
rotational can a strain -potential exist.
The line integral along any curve AS of the tangential com-
ponent of the displacement
B B
80} / q cos (q, ds) ds = I (udx + vdy + wdi)
A A
is called the circulation along the path, and for irrotational strain is
independent of the path, equal to cpB — yAj and vanishes for a closed
path.
Surfaces for which
-^=r> -=-• The normal component
Fn Fn Fn
83) Fnn = In cos (Fnri) = Xn cos(nx) -\- Yn cos (ny) -\- Zn cos (ne).
If we draw the normal in either direction from the element dS, and
if we understand by Fn the force exerted through dS by the portion
of the body lying on the side toward which n is drawn on the
portion lying on the other side, then if the normal component
Fnn = Fn cos (Fnri) is positive it is called a traction, if negative, a
pressure. In other words it is a traction if its effect is to cause the
portions of the body to approach each other, a pressure if it is to
make them recede.
The force upon any element dS can be expressed in terms of
the forces upon three mutually perpendicular plane elements at the
same point. Construct, enclosing the
point P, an infinitesimal tetrahedron
bounded by the element dS and three
planes parallel to the coordinate planes
(Fig. 146). Let the areas of the four
triangular faces be dS, dSx, dSy, dS~, the
suffix in each case denoting the direc-
tion of the normal to the face. Further
denote the stress -vector for any face by
a suffix giving the normal to that face,
and let the stress -vectors be those for
the portion of the body within the
tetrahedron. Suppose that forces are
applied to every portion of matter in
proportion to its mass, such, for instance,
as gravity, the components being X, Y, Z per unit mass. If d-c
denote the volume of the tetrahedron the X- component of these
external forces is accordingly Xgdt.
Let us now form the equations for equilibrium of the matter
contained in the tetrahedron under the influence of the external
forces and the stresses developed. The first of these is
84) Xgdr + XndS - XxdSx - XydSy - X3dSz = 0.
Fig. 146.
448 IX- DYNAMICS OF DEFORMABLE BODIES.
But since the three other sides are the projections of dS, we have
85) dSx = dS cos (nx), dSy = dS cos (ny), dSz = dS cos (nz).
Inserting these in the equation 84), dividing through by dS,
and taking the limit, as the edges of the tetrahedron become infinitely
small the ratio of the volume to the surface disappears, so that we
have finally
86) Xn = Xx cos (nx) -f X.y cos (ny) -f Xz cos (nz) ,
and similarly
Yn = Yx cos (nx) -\- Yy cos (ny) -f Y, cos (nz),
Zn = Zx cos (nx) -f- Zy cos (ny) -f Zz cos (nz).
Let us now consider the equilibrium of any portion of the body
bounded by a closed surface S. Resolving in the X- direction, we
have as the condition for equilibrium, considering both the stresses
on the surface and the volume -forces,
87)
Making use of equations 86) for Xn,
88) / / { Xx cos (nx) + Xy cos (ny) + Xz cos (ne)} dS
and by the divergence theorem, n being the outward normal,
r r ridx sx ax
89) JJJ y + w + ^
Since this must hold for every portion of the substance which is in
equilibrium, the integrand must vanish, and we have consequently
together with the result of resolving in the two other directions,
dX dX dX
These are but three of the six equations for equilibrium. The other
three are obtained by taking moments, the first being
91)
170]
NATURE OF STRESS.
449
Introducing the values of
Ynj Zn from 86) this becomes
, $Z from equations 90) and of
92) / / {y [Zx cos (nx) -f Zy cos (ny) -f Z* cos (w0)]
- 0[YX cos (w#) -f Yj, cos (ny) -f F, cos (ws)]} d$
*SL
(^
\0aj
+-*+ -
Writing the term
and
and applying the divergence theorem, all the surface integrals cancel
each other and there remains only the volume integral
93)
As before, the field of integration being arbitrary, the integrand
must vanish, and we obtain, after applying the same process to the
remaining two equations,
QA\ v 7 7 v ~y v
y±) JL z = Zyy, ZJX = jC^Zj J^y =F Lx.
We may also obtain these equations by considering the stresses
on the faces of an infinitesimal
cube (Fig. 147). We shall denote
the tangential components or
shearing stresses 94) by Tx, Ty, Tz,
the normal components or trac-
tions by PXJ Py, Pz. The stress
at any point is determined in
terms of these six components,
for we may find the stress -vector
Fn, whose direction - cosines . are
u'j P'> ?' f°r anJ stress plane
whose normal has the direction
cosines a, fl, y by equations 86),
which in our present notation
Fig. 147.
become
95)
Xn = Fna' =
Yn = Fnp =
TyK
WEBSTER, Dynamics.
29
450
IX. DYNAMICS OF DEFORMABLE BODIES.
These are the exact analogues of equations 17). In other words, the
stress -vector is a self - conjugate linear vector -function of the normal
to the stress -plane. The stress -vector Fn occupies the place of *
in 17). Accordingly the whole geometry of the linear vector func-
tion may be applied to the consideration of stress as follows.
171. Geometrical Representation of Stress. If we construct
the quadric
96) cp EEE Pxx2 + Pyy* + Pzs* + 2Txyz + 2Tyzx + 2Tzxy = ± E2
any stress -vector Fn is perpendicular to the tangent plane drawn at
the point where the normal to the stress-
plane cuts the quadric cp (Fig. 148). This
is known as Cauchy's stress -quadric. Let
its equation, referred to its principal axes,
which are known as the axes of the stress, be
97) cp = P±x2 + P2y2 + P3*2 = ± E2.
P1; P2, P3 are called the principal tractions,
being the normal stresses on the planes
perpendicular to the axes, these planes
being subject to no tangential stresses.
Thus, as for any strain we may find three
planes for which the slides vanish, so for
stress we may find three planes for which
the shearing stress vanishes.
In the reciprocal quadric,
«.2 »,2 *2
QQ\ ' — _|_ " _|_ i 7)2
the stress -vector is conjugate to its stress -plane, for the normal to
the stress -plane is parallel to the normal to cp' where it is cut by
the stress -vector. The quadric cp' is known as Lame's stress -director
quadric. In equations 17) and 14) putting Fn for -- we obtain
Fig. 148.
99)
p == _}_ ^L __ _j ±i
— pr — r2cos(?*r')
or
100)
So that the traction or component of the stress -vector normal to its
stress -plane is inversely proportional to the square of the radius-
vector of the quadric (p in the direction of that normal, or is directly
proportional to the square of the perpendicular upon the tangent
plane to the quadric cpr parallel to the stress -plane.
170, 171, 171 a] GEOMETRY OF STRESS. 451
If P±, P2, P3 are all of the same sign the quadrics cp and 9?'
are ellipsoids. If they are positive we must take the positive sign
with R?, and the normal stress on every plane is a traction. If they
are negative, we must take the negative sign, and the normal stress
is always a pressure. If one of the P's has a different sign from
the two others, we use both signs and have pairs of conjugate
hyperboloids. In this case for directions parallel to the generators
of the asymptotic cone cp = 0 to the stress quadric, we have r infinite
and Fnn = 0. Accordingly for stress -planes perpendicular to these
generators, the normal stress vanishes or the stress is a shearing
stress. These planes envelop a cone called Lame's shear -cone, which
divides the directions for which the normal stress is a traction from
those for which it is a pressure.
In the reciprocal quadric qp', when the radius vector is infinite,
it lies in its conjugate plane, the stress -plane. But the radius vector
to this quadric has the direction of the stress -vector, so that the
shear -cone is the asymptotic cone to this quadric that the stress -vector for any
plane is directly proportional to the radius vector in its own direction.
This ellipsoid is called Lame's stress -ellipsoid, or ellipsoid of elasticity.
171 a. Simple Stresses. A simple stress is one that contains
but a single constant in its specification. These are:
1°. Uniform traction or pressure.
P~P ~P T>
i = -L a = J-n - JC .
All the quadrics are spheres and every stress is normal to its plane
and of invariable amount P. Such a stress is physically realized by
a body subjected to hydrostatic pressure.
29*
452 IX. DYNAMICS OF DEFOBMABLE BODIES.
2°. Simple traction ,
PX=P, Py = P. = 0,
105) T T T
J-x— J-y= -Lz — M-
The stress quadric is
106) y = Px2 = ± 1,
a pair of planes perpendicular to the X-axis at a distance from
the origin. The stress on any plane is parallel to the X-axis. The
stress -director quadric and the shear -cone reduce to the axis of X,
all planes tangent to which experience only shear.
Cauchy's ellipsoid,
107) P V + 0 • f + 0 - £ = 1,
with axes, p» oo, oo, is a pair of planes perpendicular to the X-axis,
and Lame's ellipsoid with axes, P, 0, 0, becomes simply that part of
the axis of X from x = — P to x = P. From the property of this
ellipsoid the stress -vector is proportional to the perpendicular on the
tangent plane parallel to the stress plane. Since the tangent plane
here always passes through one of the extremities we have
108) , Fn = Pcos(nx)
as is indeed evident from equations 95).
3°. Simple shearing stress.
p _ p — p — o
, y * '
Tz = T, Tx = Ty = 0.
Equations 95) become
110)
The stress quadric is
111)
which represents a pair of rectangular hyperbolic cylinders with the
semi -axes — =• The stress -director quadric is
yr
112)
The shear cone xy = 0 represents the coordinate planes of X.
and YZ.
171 a] SIMPLE STRESSES. 453
A shearing stress may also be written, referred to its principal axes,
113) / = ~ 'T ' "
J.X= -Ly= J-Z = V,
when the stress quadric becomes
the pair of hyperbolic cylinders referred to their axes and
Xn = Fna' = Pa,
-J -J £ \ T7* 77* /? ' ~P ft
np ~J71 f f\
We accordingly have
= or
/?'
that is, all stresses are parallel to the XY- plane, and the stress-
vector and the projection on the XY- plane of the normal to the
stress plane make equal angles with
the X-axis on opposite sides (Fig. 149).
Squaring equations 115) and adding,
116) Fn* = P2(a2 + /32)=P2(1-^2).
If y = 0, that is, if the plane is tangent
to the Z-axis
Fig. 149.
the normal stress being a traction if
the normal to the stress -plane falls
nearer to the X-axis, a pressure if nearer to the F-axis.
The shear cone x2 — y* = 0 is composed of the two planes
bisecting the dihedral angle between the XZ- and YZ- planes.
From this manner of representing the stress it is evident that a
simple shearing stress is equivalent to an equal traction and pressure
in two directions perpendicular to each other. Compare the repre-
sentation of a shearing strain as an equal stretch and squeeze. For
this case Lame's ellipsoid -=, (x2 -f y2) + — = 1 has the axes P, P, 0
and reduces to a circular disc normal to the ^-axis. Since all tangent
planes pass through its edge,
117)
as abore in 116).
Fn = P sin (nz) =
454 IX. DYNAMICS OF DEFORMABLE BODIES.
172. Work of Stress in producing1 Strain. If every point
in a body move a distance dq, whose components are 8ti, dv, dw,
and if there act upon every unit of mass of the body the external
forces X, Y, Z, and upon each unit of surface the forces Xn, Yn, Zn,
the work done by all the forces in the displacement is
118) dW= I UXndu + Yndv + Zndw}d8
+ / / I Q[Xdu + Ydv + Zdw}dr,
which becomes by equations 86),
119) d W = I j {[_XX cos (nx) + Xy cos (ny) + Xz cos (ngj] du
+ [Yx cos (nx) + Yy cos (ny) + Yz cos (nz)\ dv
+ [Zx cos (wfl?) 4- Zy cos (wy) + ^ cos (mi)] dw} dS
and transforming surface integrals into volume integrals by differentia-
tion in the manner of the divergence theorem and making use of
equations 94),
YydV
w + F^i; + Zdw)
•"»
az 0Zf, dZ
' ' + ^
+ X^+Y1r^ + Z,
TT /^^'^ _j_ ^V\ , ^ (W<* .
~\ J-z \ "iaTT ' Q * / ~t~ ^a; I o«
172, 173J WORK OF STRESS. 455
By equations 90) the coefficients of du, dv, dw vanish identically, so
that, interchanging the order of differentiation and variation,
121) *W-{x.*& + Y,* + Z.9% + Y,
or in our later terminology,
122) 8W= j i C{Px8sx + Pydsy + P,8s, + 2Txdgx
+ ZTydgy + 2T2dg2}d>t.
Thus each of the six components of the stress does work on the
corresponding component of the strain, and the work per unit volume
in any infinitesimal strain is the sum of each stress component by
the corresponding strain produced, except that with our terminology
the shearing stresses are multiplied by twice the shearing strains or
slides.
173. Relations between Stress and Strain. If a body is
perfectly elastic the stresses at any point at any time depend simply
upon the strain at the point at the time in question, so that if the
elastic properties of the body are known at every point the stress
components will be known functions of the strain components, which
may differ from one point of the body to another. The stresses will
be uniform and continuous functions of the strains and may be
developed by Taylor's theorem. If then the strains are small, the
terms of the lowest orders will be the most important. The strains
dealt with by the ordinary theory of elasticity are so small that it
is customary to neglect all terms above those of the first order. The
results thus obtained are in good accordance with those obtained by
experiment under the proper limitations. The law that for small
strains the stresses are linear functions of the strains may be regarded
as an extension of the law announced in 1676 by Hooke in the form
of an anagram,
ce^^^nosssttuu
Ut Tensio sic Vis.
The force varies as the stretch, or in our terminology the stress
varies as the strain. Making this assumption we accordingly have
X = 9>oi
456 IX. DYNAMICS OF DEFORMABLE BODIES.
The gp's will in general be functions of the coordinates of the point,
but if the body is homogeneous, that is alike at all points, they will
be constants. We shall assume this to be true. If there be a natural
state of the body or one in which the body is in equilibrium under
the action of external forces, so that the stresses vanish for this
state, it is convenient to measure the strains from the natural state.
Then the stresses and strains vanish together, so that the terms
^P01, . . . qpog vanish. For such a body there are accordingly thirty -six
constants qp, the so-called coefficients of elasticity. In the case of a
gas there is no natural state, for a gas is never in equilibrium,
unless kept so by an envelope, so that every portion of the gas
always experiences pressure, consequently we cannot measure the
strains from any natural state.
We have now the theory of elasticity as it was left by its
founders, Navier and Cauchy. The idea is due to Green1) of supposing
the elastic forces to be conservative and accordingly due to an energy
function of the strains. If we call the function &(sx> sy, sz, gx, gy, gz]
we have for the total potential energy due to any strain
124)
The work done in changing the strain is then
125)
Comparing this with equations 121 — 122) we find
If then the stresses are to be linear functions of the strains,
£> must be a quadratic function, and, if we measure from the natural
state, a homogeneous quadratic function. A homogeneous quadratic
function of six variables contains twenty -one terms, so that instead
of thirty -six elastic constants for the general homogeneous body we
have only twenty -one, that is, the determinant of the qp's in equa-
tions 123) is symmetrical, fifteen coefficients on one side of the
1) Green, Mathematical Papers, p. 243.
173, 174] STRESS -POTENTIAL. 457
principal diagonal being equal to the corresponding fifteen on the
other side.
If the body besides being homogeneous is isotropic, that is, at
any point its properties are the same with respect to all directions,
there are many relations between the coefficients, so that the number
of independent constants is much reduced. In an anisotropic or eolo-
tropic body there are generally certain directions (the same for all
parts of the body) with reference to which there is a certain symmetry,
so that there are various relations involving a reduction in the number
of constants. Such bodies are known as crystals. We shall deal
here only with isotropic bodies.
174. Energy Function for Isotropic Bodies. In isotropic
bodies the stresses developed depend only on the magnitude of the
strains, not on their absolute directions with respect to the body.
Accordingly if we change the axes of coordinates the expression for
the energy must remain unchanged, or the energy function is an
invariant for a change of axes. The cubic for the axes of the
elongation quadric 58) belonging to the shift -equations 50) is the
determinant
— A, g, , gy
g, , sy - I, gx
9y > 9* , S* —
or expanding the determinant,
128) 43 - (sx + sy + s.) tf 4- (sysz + szsx + sxsy — gl — g} — gl) I
+ sxgl + sygl + s,gl - sxsys, - 2gxg1Jgz = 0.
If the roots are A17 Z2? A3, the equation is
129) tf - (^ + A2 + *3) tf + (M2 + Vs + Mi) * - M2*3 = 0.
If we transform to another set of axes X'Y'Z' with the same origin,
so that the strain components are sx', sy', sz; gxr, gy; gz', since the
elongation quadric is a definite surface, the equation for its axes
must have the same roots as before. Accordingly its coefficients are
invariants. The roots A1? A2, A3 are the stretches for the directions
of the principal axes of the strain. Therefore we have the three
strain invariants, symmetrical functions of the roots,
/! = ^ + ^2 + Ag = Sx + Sy + 8,,
130) I2 = A^2 + M3 + Vi = svsz + B*SX + sxsy -gl-gl — gl,
J3 = ^A^g = 2gxgyg, -f sxsys, - sxgl - svg*y — s,g*,.
The invariant Jx represents the cubical dilatation (?, which by its
geometrical definition is evidently independent of the choice of axes.
127)
0,
458 IX- DYNAMICS OF DEFORMABLE BODIES.
The energy function for an isotropic body, being unchanged when
we change the axes, can contain the strains only in the combinations
I,, J2, I3, but these are of the first, second and third degrees
respectively, and since <& is of only the secoud degree it cannot
contain J3.
Since it is homogeneous (except for a gas) it can contain Jt
only through its square. We therefore have
131) 0 = - PIi + AI* + #/2,
where P, A, B are constants. P is zero, except for gases, and is then
positive, for if the gas expands it loses energy. The constant A
refers to a property common to all bodies, namely, resistance to
compression, and is positive, for work must be done to compress a
body. The constant B is peculiar to solids.
All symmetrical functions of the roots may be expressed in terms
of the invariants, for example:
132) &-*,)« + ft -*»)' + &- AJ1
= 2 ( v + v + V) - 2 ft A2 + 1, ^ -t- A3 ;g
A2 + A3)2 - 6 ft;, + A2*3 + A,^)
Also
2&Jt + J,J, + J,i1)~(i1 + l1 + ^'-.(V+V + V), or
r2
We may accordingly write A 1^ -f BI2 as a linear function, of J,2
and of either (^ - A2)2 + (A2 - A3)2 + (X - Aj2, or of V + V + A32.
Suppose we write the quadratic terms
134)
y2 + (is - Itf + (i, -
which is the form given by Helmholtz. The constant H, being
multiplied by tf2, refers to changes of volume without changes of
form, representing in this case the whole energy, for if there is no
change of form the stretches of the principal axes, ^, A2, A3 are equal.
The term in C on the other hand refers to changes of form without
change in volume, for it vanishes when A, = A2 = ^3, and represents
the whole energy if = 0. A perfect fluid is defined as a body in
which changes of form produce no stress, so that for such bodies
C = 0.
We may also write
135) A I* + -BZ, - -STCV + V + V) + K& & + I, + Itf,
174] ISOTBOFIS^]^Bpi1^gJg^TIAL. 459
which is the form used by Kirchhoff. We have then for the rela-
tions between the constants,
136) H-C = 2K® and 3C=2K
or
(1 \ 9
& _i_ _\ n — JC
3 / 3
Accordingly for liquids in order to have (7 = 0 but H finite, we
must put K= 0 and ® = oo, so that 2K® = J7.
Now since
J.O < ) A< ~j~ A2 ~j" Ag === -tj ^ -*-2
f _L __L ^2 O ' /" j_ i_ 2 ^2 2\
we have for solids, liquids and gases,
138) ® = K®0* + K{(s* + sl + sl)
We shall make use of the more common notation
139) 2K& = 1, K=ii.
(Thomson and Tait make use of the constants k for H and n for /i.)
We have accordingly
140) $ = ±Ji(sx + sv + s,y 4- ii{sl + s2 -f si + 2(^2 + ^ + <72)}
- P (S,,. + Sy + S,).
The constant ^ like (7 refers to changes of form and vanishes for
perfect fluids. In the present notation by equations 136) we have
141) H = ^ + l,
so that both >L and p are involved in changes of volume. We thus
see that isotropic bodies possess two elastic constants. By means of
certain assumptions as to the nature of elastic stresses, making them
depend upon actions between molecules, Cauchy and the earlier
writers on elasticity reduced the energy function to a form depending
on a single elastic constant, the same theory reducing the number
of constants for an eolotropic body from twenty- one to fifteen. For
this theory the reader may consult Neumann, Theorie der Elastizitat,
Todhunter and Pearson, History of the Theory of Elasticity. Experiments
have not however confirmed this theory, and it is no longer generally
held to be sound. Thomson and Tait inveigh against it with particular
emphasis. We shall accordingly assume that an isotropic body has
two independent constants of elasticity A and ^.
460 IX. DYNAMICS OF DEFORMABLE BODIES.
175. Stresses in Isotropic Bodies. We may now calculate
the stresses by means of equations 126), inserting the values of
Sx, sy, ss, gx, gy, gz.
.
v ' 1 d$ ftw dv\
J-2 = AV = IT o — = V* o -- h -o—
2 C9X \dy o*J
1 fo du . dw
The first equation of equilibrium 90) becomes
i^o\
143)
or, considering the value of tf, we may write the equation with its
two companions,
144)
The equations at the surface of the body are by 86), using the
above values of the stresses,
Xn = (it + 2{*|| - P) cos (na?)
f ^ (du . dw\ f N
cos (w^) ^. ffJiw-t W cos (w^'
c°s (nx^ + (ie + ^ ll - p) cos ^
146)
. dw ,
*=~-tl- + cos v + cos
-r P cos (ne).
175, 176] STRESSES IN ISOTROPIC BODIES. 461
176. Physical Meaning of the Constants. Let us consider
a few simple cases of equilibrium with homogeneous strain under
stress, there being no impressed bodily forces X, Y, Z, and putting
P = 0.
1°. A simple dilatation.
u = ax, v = ay, w = as,
sx = a, sy = a, sz = a, gx = gy = gz = 0,
& = sx + sy -f sz = 3a,
146) Xx = Yy = Zz
The surface forces become simply
Xn = pcos(nx),
147) Yn=pco*(ny),
Zn =pQos (n0),
or the surface force is normal to the surface, and
148) F,, = P
The ratio of normal traction to cubical dilatation, or of normal
pressure to cubical compression,
149) ^, = n±^ = fi
is called the bulk -modulus of elasticity, the term modulus being
applied in general to the ratio of the stress to the strain thereby
produced.
2°. A simple shear.
u = ay, v = w = 0,
sx = sy = sz = (3 = gx = gy = 0,
150) , = , Xx = Yy = Z2 = 0,
= Z, = ZX = X3 = 0,
ny), Yn=Tcos(nx),
151) Fn = Tycos2(nx) + cos2 (ny) = Tsm(nz}, as in 117).
462 IX- DYNAMICS OF DEFORMABLE BODIES.
The ratio of the tangential stresses on the XZ and YZ planes
to the amount of the shear produced ,
is called the sliear modulus of elasticity, or the simple rigidity.1)
3°. A stretch -squeeze.
u = ax, v = — ~by, w = — bz,
153) sx = a, sy = sz = — l}, <5 = a — 26,
Xx = I (a — 26)
Xn = {A (a — 26) + 2^#}cos (nx),
Yn = {A (a - 26) - 2/^6} cos
If we choose a and 6 so that
155> 6 =
we have
Xn = 2 11 (a + 6) cos (nx) = ~ • a cos (nx).
If the body is a cylinder, with generators parallel to the X-axis;
bounded by perpendicular ends, experiencing a normal traction p,
there is no force on the cylindrical surface, for which cos (nx) = 0,
and on the ends
157)
The ratio of the tractive force to the -stretch a,
E=^-
a
may be called the stretch or elongation modulus, and is generally
known as Young's modulus.
The ratio of the lateral contraction to the longitudinal extension
159) r, = ± l
1 a
2 IL
1) Thomson and Tait use the notation: bulk -modulus =jfe = ^, -[--—, simple
rigidity =w, m = ^-(-/i = — -J-^-
8
176, 177] ELASTIC MODULI. 463
is called Poisson's ratio. According to Poisson and the older writers
A = ^, so that t] = — - We must certainly have
- 1< — y ^ < 0, making the rigidity negative. If 17 < — 1,
making the bulk -modulus negative. No known bodies have 77 < 0,
and in experiments on isotropic bodies 77 has generally been found
nearly equal to —> the value assumed by Poisson, the value being
found to approach more nearly to Poisson's value the more pains
were taken the secure isotropic specimens.
The bulk-, shear- and stretch -moduli and Poisson's ratio are the
important elastic constants for an isotropic body, any two of which
being known, all are known.
CHAPTER X.
STATICS OF DEFORMABLE BODIES.
177. Hydrostatics. Let us now consider the statics of a perfect
fluid, that is, a body for which p = 0. If each element of the fluid
is subjected to forces whose components are X, Yf Z per unit mass,
equations 144), § 175 reduce to
while the equations for the surface forces 145) become
Xn = (JL6 — P) cos (nx),
2) rn
The surface force is accordingly normal and equal to
3) ll times the derivative
of 6 by the corresponding derivative of (— p), our equations of
^equilibrium 1) are
^ ~ dx' ^ dy' ^ dz
Thus the fluid can be in equilibrium only under the action of bodily
forces of such a nature that Q times the resultant force per unit
mass, that is to say, the force per unit volume, is a lamellar vector.
If the pressure at any point depends only on the density, and
conversely, and we put — = -j— '»
5)
«/ *
so that
dP _ dP dp _ 1 dp
dx dp dx Q dx
6) d^=dPdp==^3p)
dy dp dy Q dy'
dP _ dP dp _ 1 dp
dz dp dz Q dz
7,
Our equations 4) are
dP dP dP
Accordingly in this case the bodily forces per unit mass must be
conservative. If V is their potential, multiplying equations 4) by
dXj dy, dz respectively and adding, we have
8) Q (Xdx + Ydy + Zde) =*-gdV
If we have two fluids of different densities in contact we have at
their common surface
9)
so that
10) (ft -
therefore dV and dp are each equal to zero and the surface of
separation is a surface of constant potential and constant pressure.
1) Not the constant P in 3).
177, 178] EQUATIONS OF HYDROSTATICS. 465
Also, since, by 8) V differs from — P only by a constant, the sur-
faces of equal pressure are equipotential or level surfaces. If the
fluid is incompressible y is constant, so that we have
11) -V= const. + -•'
For gravity we have, if the axis of Z is measured vertically upward
so that
12) P~tt(C
If, neglecting the atmospheric pressure, we measure z from the level
surface of no pressure
13) P = -9Q*,
which is the fundamental theorem for liquids, namely, that the pressure
is proportional to the depth.
178. Height of the Atmosphere. If we consider a gas whose
temperature is constant throughout, the relation between the pressure
and volume is given by the law of. Boyle and Mariotte
p = a$,
accordingly
14) P = f*2 = r°±!L = a log e + const.
and
15) V=gs = c — a
16) ? = 9o«",
where p0 is the density when 0 = 0.
Thus as we ascend to heights which are in arithmetical pro-
gression, the density decreases in geometrical progression, vanishing
only for # = oo. If on the other hand, we consider the relation
between pressure and density to be that pertaining to adiabatic
compression, that is compression in such a manner that the heat
generated remains in the portion of the gas where it is generated,
we shall obtain a law of equilibrium corresponding to what is known
as convective equilibrium. The temperature then varies as we go
upward in such a way that, if a portion of air is hotter than the
stratum in which it lies, it will rise expanding and cooling at the
same time until its temperature and density are the same as those
of a higher layer. When there is no tendency for any portion of
WEBSTER, Dynamics. 30
466 X. STATICS OF DEFORMABLE BODIES.
air to change its place convective equilibrium is established. The
principles of thermodynamics give us the relation for adiabatic com-
pression
17) p = !)(>*,
where % is the ratio of the specific heat at constant pressure to that
at constant volume, whose numerical value is about 1.4. We then have
18)
Since x > 1, p diminishes as 8 increases and is equal to zero when
gz = cf so that on this hypothesis the atmosphere has an upper limit,
which may be calculated when the value of Q for a single value of 8
is known.
It is obviously improper to consider the equilibrium of the
atmosphere to an infinite distance without taking account of the
variation of gravity as the distance above the surface of the earth
increases. Considering the earth to be a sphere with a density that
is a function only of the distance from the center, we have^ with 7
positive1), as in §§ 123, 149, instead of equation 8),
Xdx+
so that on the hypothesis of equal temperature
19) - - — = const. — a log p,
20) 9 = 9,e^.
On this hypothesis the density decreases as we leave the earth, but
not so fast on account of the diminution of gravity, so that at infinity
the density is not zero but equal to the constant
In this example we have neglected the attraction on the gas of
those layers lying below. From equation 20) the barometric formu
is obtained.
Proceeding in the same manner for convective equilibrium, we have
21) -
-
Here again Q decreases as r increases, giving an upper limit to the
atmosphere for Q = 0 for a finite value of r.
1) It is to be observed that in equation 8) the forces are taken as the
negative derivatives of the potential, but as in the following examples involving
the earth's attraction they are obtained by multiplying ilae positive derivative byy,
we must in the integral equation change the sign of V and multiply by y.
178, 179] ATMOSPHERE. ROTATING FLUID. 467
179. Rotating Mass of Fluid. If a mass of fluid rotates
about an axis with a constant angular velocity ro, we may by the
principle of § 104 treat the problem of motion like a statical problem,
provided we apply to each particle a force equal to the centrifugal
force. If we take the axis of rotation for the ^-axis the centrifugal
force may be derived from the potential,
If an incompressible liquid rotates about a vertical axis and is under
the influence of gravity, we have by 11),
23) V=gz- y02 + r%
24) p = o(c-gs-\- ^(a^ + y*)).
V * /
Consequently the surfaces of equal pressure are paraboloids of revolution.
Measuring 0 from the vertex the equation of the free surface, for
which p = 0, is
O ^ \ ^ / 2 i 2\
The latus rectum is \> On this principle centrifugal speed indicators
are constructed.
An important case which we have already treated by this method
in § 149 is the shape of the surface of the ocean. If we seek an
approximation, assuming the earth to be centrobaric, the potential
due to the attraction of the earth and centrifugal force will be, as
we find either directly, or by putting K=Q in § 149, 140),
and the equation to the surface of the sea will be, U= const., which
may be written, writing — for the constant,
where ty is the geocentric latitude, and a is the polar radius. In the
case of the earth ,.. = noo . . which is so small that the second
ym. 288.41
term may be considered small with respect to the first, and its square
neglected. Accordingly putting in this term r = a the equation of
the surface is
f w2as 1
1 ' 2yM ' ^J
30!
468 X. STATICS OF DEFORMABLE BODIES.
which is the equation of an oblate ellipsoid of revolution. The
ellipticity is
> + ££)- Sir'
The difference between this and the value — - given in § 149 is due
to the fact that we have neglected the attraction of the water for
itself and that the nucleus is not exactly centrobaric.
18O. Gravitating, rotating Fluid. A problem of great
importance in connection with the figure of the earth and other
planets is the form of the bounding surface of a mass of homogeneous
rotating liquid under the action of its own gravitation.
If V is the potential of the mass of fluid at any internal point,
and we take the X-axis for the axis of rotation, we have
The form of the function V depends upon the shape of the bounding
surface of the liquid, which is to be determined by the problem
itself. The complete problem is thus one of very great difficulty
and has been only partially solved.1)
We will examine whether an ellipsoid is a possible figure of
equilibrium.
We have found in § 157, 37) for the potential of a homogeneous
ellipsoid
QO
C '( T2 it2 ?"*• \ fiii
30) V=x<>al>c J |l--^-&^--^| ™ —
o
= const. -^{Lx2 + Mf +
where
du
L = 2itoal>c I -
J (a^-\-u}-[/(a^-\-u)(l^-^
31) M=2it(>abc C =
I (h% _j_ -jA I/Yd2 -4- u
^/ \ J r \
0
I -
J (
1) Poincare, "Sur Tequilibre (Tune masse fluide animee d'un mouvement
de rotation." Acta math., t. VII, 1885. Also, Figures d'equilibre d'une masse
fluide. Paris, 1903.
179, 180] GRAVITATING ROTATING FLUID. 469
Inserting in the integral equation 11) (see footnote p. 466),
32) ~ + i{ Lx* + W + N^ - 7 (f + *2) } = c°™t.,
the surfaces of equal pressure are similar to the ellipsoid
33) L,
and ^0) = 0 for I = 2.5293,
for which value
470 x STATICS OF DEFORMABLE BODIES.
The course of the function
is shown in Fig. 150, from which
it is evident that if
< 0.22467
Fig. 150.
there are two values of I satisfying
equation 38) and accordingly two
possible ellipsoids of rotation. If
on the contrary
> 0.22467
"
no possible ellipsoid of rotation is a figure of equilibrium.
When co is very small one of the values of 'L tends to zero and
the other to infinity, that is, one of the ellipsoids is a sphere, the
other a thin disc of infinite radius.
In the case of the earth using the value of yQ of § 123 and
of co of § 149. - - = .00230. and the smaller of the two values
7 %icyg
of A coincides most nearly with the actual facts1), giving
A2 =0.008688, e = ^-
The actual ellipticity being however ^^ we can only conclude that
^jy y
the earth when in its fluid state was not homogeneous.
The transcendental equation 36) written out is
40)
HD
-e2) C\-*-
>J la'+«
or otherwise
41) (V -
du
{<*»*+
=0.
Besides the solution 6 = c there is another given by putting the
integral equal to zero. When a = 0, the integrand and consequently
the definite integral is negative, when a = — - - the integral is
j/&2-t-e2
positive. There is accordingly a real value of a which satisfies the
equation and there is an ellipsoid with three unequal axes which is
a possible figure of equilibrium, if co lies below a certain limit.
This result was given by Jacobi in 1834. For further information
on this subject the reader is referred to Thomson and Tait, Natural
Philosophy, §§ 771—778.
1) Tisserand, Traite de Mecanique Celeste, Tom. II, p. 91.
180, 181] POSSIBLE ELLIPSOIDS. 471
181. Equilibrium of Floating Body. Let us apply the equa-
tions of equilibrium to a solid body immersed in a fluid under the
action of any forces. Let us find the resultant force and moment
of the pressure exerted by the liquid on the surface of the body.
If we call the components of the resultant H9 H, Z, and of the
moment L} M, N, we have
S = / / p cos (nx) dS,
42) H = I I p cos (ny) dS,
Z= I I pcoB(ne)dS,
L = I I [yp cos (ns) — zp cos (ny)} dS,
43) M = / / {#p cos (nx) — xp cos (nz)} dS,
N = I I {xp cos (ny) — yp cos (nx)} dS.
If the body is in equilibrium it is evident that we may replace it by
the fluid which it displaces, which would then be in equilibrium
according to equations 4), and might then be solidified without disturb-
ing the equilibrium.
If the body is only partly immersed we must apply the integration
to the volume bounded by the wet surface and a horizontal plane
forming a continuation of the free surface of the liquid and called
the plane of flotation. Over this plane p = 0, consequently the surface
integral is taken only over the wet surface, while the volume integral
is as before taken over the volume of the fluid displaced. With this
understanding we may couvert the surface integrals into volume
integrals taken throughout the space occupied by the displaced liquid,
that is, within the surface of the solid body below the plane of
flotation. We thus have
441
472 X. STATICS OF DEFORMABLE BODIES.
45)
If the force acting on the liquid be gravity we have
X = Y=0, Z=-g,
accordingly
where m is the mass of the fluid displaced by the body. This is the
Principle of Archimedes: A body immersed in a liquid has its weight
diminished by the weight of the displaced liquid.
For the moments we have
L
46)
M
= 9 i I I Qydt = gmy,
= — g I I I Qxdr = — gmx,
N = 0,
where x, y denote the coordinates of the center of mass of the
displaced liquid. If the body is in equilibrium, by Archimedes'
Principle the weight and therefore the mass of the body is equal to
that of the displaced liquid. Consequently the resultant of the forces
acting on the body is equivalent to a couple whose members are
forces mg exerted downward at the center of mass of the body and
upward at the center of the mass of the displaced liquid. If the
couple is to vanish one of these must be vertically above the other.
The center of mass of the displaced liquid is called the center of
buoyancy of the body.
If the floating body is slightly displaced through a small angle
deo from the position of equilibrium by the application of a couple,
the mass of the displaced fluid must remain unchanged, but the
position of the center of buoyancy is slightly altered. Let us take
the origin in the intersection of the old and new planes of flotation
(Fig. 151). For the new position the figure is to be tilted in the direction
of the arrow until the new position of the water line W'L' is horizontal.
The old center of buoyancy JB is now no longer under the center
of mass Gr and consequently, if the same portion were immersed,
181] FLOATING BODY. 473
the body would be acted upon by a couple equal to mg times the
horizontal distance between the verticals through G and B, and tending
to increase the angular dis-
placement. If 1) denote the ^ M
length GB which now makes
an angle d co with the vertical,
this horizontal distance is w
, and the couple mgbdco.
\
j-
Ju
7 _ . j.-, — i>— _L-^" \ """^-r'
But this is not the only couple, \ B )
for the immersed part is now \^__ ^
different from that formerly Fig 151
immersed by the volume of
the two wedges of small angle tfco, the wedge of immersion EOE1
and the wedge of emersion DOD'. The buoyancy added by the wedge
of immersion and that lost by the wedge of emersion both produce
moments in the direction tending to decrease the displacement. These
moments must accordingly be subtracted from that previously found,
to obtain the whole moment tending to overset the body.
It is evident that if no vertical force is to be generated, the
volumes of the wedges of immersion and emersion must be equal.
Since the wedges are infinitely thin we may take for the element
of volume
at = zdxdy = yo&dxdy.
The condition for equal volumes is then
/ I zdxdy = dG> I I
47)
the integral being taken over the area of the plane of flotation. This
will be the case if the axis taken through the center of mass of the
area of flotation. The moment due to the wedges is
48) L'=g$ I I j ydx = gQ I j
sydxdy
/ /
y*dxdy =
where KX is the radius of gyration of the area of flotation about the
X-axis.
In like manner
49) M' = — gg I I I xdt = — g$ I I xzdxdy
= — ggdco I I xydxdy.
474 X- STATICS OF DEFOEMABLE BODIES.
If we take for the axes of x and y the principal axes of the area
of notation the integral / / xydxdy vanishes. Accordingly a rota-
tion about a principal axis through the center of mass of the plane
of notation developes a couple about that axis of magnitude ggdoSnl
tending to right the body. We have accordingly for the whole
moment of the righting couple
50) L = gdco (p£x| — mV).
On account of the change in the immersed part the center of
buoyancy has moved from B to B'. If we draw a vertical in the
new position through B\ the point M in which it cuts the line BG
is called the metacenter and the distance MG = h, the metacentric
height. Since the couple acting on the body is composed of the two
forces, mg acting downward at G and upward at B', it is evident
that if the equilibrium is stable or the righting couple is positive M
must be above G. The arm of the couple being the horizontal
projection of MG is equal to hx-d(o9 and L = mghxd(D. We
accordingly have, inserting this value of L in equation 50) for the
metacentric height
51) mhx
dividing by m and writing — = V, the volume of the displaced liquid,
52) k_£j?_6.
The equilibrium is stable or unstable according as this is positive or
negative.
For the displacement about the F-axis we have in like manner
a couple proportional to the displacement, with a new metacentric
53) ft*=y-&,
where Ky is the radius of gyration of the plane of notation about
the F-axis. It is evident that the metacentric height is greater for
the displacement about the shorter principal axis of the section. Thus
it is easier to roll a ship than to tip it endwise.
Since the rotation about either axis is resisted by a couple
proportional to the angular displacement, the body will perform
small harmonic oscillations about the principal axes with the periodic
times «««
and !
where Kx and Ky are respectively the radii of gyration of the solid
about the principal axes in the plane of notation.
181, 182] METACENTER. 475
Introducing the values of hx and hy we may write
54) '£
Since for a body of the shape of a ship 5T and # increase together,
we see that the larger x corresponds to the shorter time. The
pitching of a ship takes place more rapidly than the rolling.
The locus of the center of huoyancy for all possible displacements
is called the surface of buoyancy, and the two metacenters are the
centers of curvature of its principal sections. Evidently the body
moves as if its surface of buoyancy rolled without friction on a
horizontal plane, for it would then be acted on by the same couple
as under the actual circumstances.
182. Solid hollow Sphere and Cylinder under Pressure.
We have dealt in § 176 with a few cases of equilibrium of solid
bodies under stress, where the strain produced was homogeneous.
We shall now treat a number of cases in which the strain is not
homogeneous. If there are no bodily forces the equations 144), § 175,
become
55)
Forming the divergence, by differentiating respectively by x, y, 2,
and adding, and interchanging the order of the operations d and A,
56) (I + 2ii) 4 = ^, ™ = 87; *-^ = «»
and (p is equal to the potential of a mass of density - — > coinciding
4 7t
with the body under investigation. If this be a sphere or spherical
shell, we find, as in §§ 125, 135,
er>\
58)
59) 9^^+,
where & is a second arbitrary constant, from which we have
476 x- STATICS OF DEFORMABLE BODIES.
60) c_?
dy
flz la*" ,,.2) „. ;
a b
dw _ 36^o? du- _ Sbzy div _ a b
giving (? = a. The values of the surface forces are by 145), § 1757
I~« ^ /a b . 3&^2\~| f ^
Xn = [id + 2/i (¥ - p + -p-)J cos (nx)
r / N /NT
/ cos * + ^ cos (
# cos Wj8f + ^ cos
H — ~ [^ cos (Mic) 4- y cos (wt/)].
Collecting the terms
bx Ix 11 / \ z / \\ fttibx Qiib
p- (7 cos (ns) + f cos (ny) -f 7 cos (»*)] = -^- = ^- cos
and writing
63)
these become
64) Xn=pcos(nx), Tn = pcos(ny^ Zn=pcos(nz),
so that p is a normal traction or pressure. If E± and E2 are the
internal and external radii of the shell,
pi = Ha + jfsr'
which determine a and 6.
_
66) ^8 s
182] HOLLOW SPHERE AND CYLINDEE. 477
In Oersted's piezometer the internal and external pressures are equal,
so that 6 = 0, and the sphere receives a homogeneous strain, which
is of the same magnitude, 6 = ^~> as if the sphere were solid. A
second practical application is found in the correction of thermometers
due to the pressure of the mercury causing the bulb to expand, the
amount of expansion being found from 66).
In treating the case of a very long hollow cylinder, we proceed
in precisely the same manner, except that the problem is a two
dimensional one. We will number the equations in the same manner,
with the addition of an accent. The formulae have an application
in finding the pressure able to be borne by tubes and boilers.
57') w = ' V==-' w==0> 6
59')
d fp (a . , b \ x
u = -^~ = ( — r + - ) -
dx \2 r/r
dm /a , b\ y
v = -2- = ( — r + — I—
dy \2 r) r
du a b 2bxz du 2bxy
d^Y + r*" ~r^' J~y = ~^~
dv _ 2bxy dv a b
Jx = ~7^"; ~d = ¥ + T2
•v fi o (a . b 2bx*\~\ , x 4:bxy
Xn = \la + 2fi (- + T2 -- -H cos (nx) -- -f- cos (ny),
62'")
' ^ ±bxy , r, 0 la . b 2bx*\~] , ^
Yn = - -^ cos nx + \la + 2^ (- + -2 - -^jj cos (ny),
63')' l^Ol+fOa-^'
64f) Xw = p cos (wa?), TJ, =._p cos (ny),
66')
«-,-
_2 f~— —\
478 X. STATICS OF DEFORMABLE BODIES.
183. Problem of de Saint -Veiiant. We shall now treat a
problem dealt with by the distinguished elastician Barre de Saint-
Venant, in two celebrated memoirs on the torsion and flexion of
prims, published in 1855 and 1856.1) For the full treatment of this
subject the reader is referred to Clebsch, Theorie der Elasticitdt, or
especially to the French translation of the same work, edited with
notes by de Saint -Venant himself. The problem is thus defined by
Clebsch, whose analysis is here followed.
What are the circumstances of equilibrium of a cylindrical body,
on whose cylindrical surface no forces act, and whose interior is not
subjected to external forces, under which the longitudinal fibres
composing the body experience no sidewise pressure?2) What forces
must act on the free end surfaces, in order to bring about such
circumstances?
We have already treated a special case of this problem in 3°,
§ 176. If the ^-axis be taken parallel to the generators of the
cylindrical surface of the body, the conditions that adjacent fibres
exercise no stress on each other perpendicular to their length are
67)
The conditions at the cylindrical surface are
Xn = Xx cos (nx) H- Xy cos (ny) + Xz cos (nz) = 0,
68) Yn = Tx cos (nx) + Yy cos (ny) + Yz cos (ne) = 0,
Zn = Zx cos (nx) + Zy cos (ny) + Zz cos (nz) = 0,
of which the first two are satisfied identically, since cos nz = 0, and
the last is simply,
69) . Zx cos (nx) + Zy cos (ny) = 0.
In order to remove the possibility of a displacement of the
cylinder like a rigid body involving six freedoms we will suppose
fixed: a point, a line, and an infinitesimal element of surface of one
orthogonal end or cross -section. Take the fixed point as origin, so
that UQ '= v0 = WQ = 0. Take the fixed element of surface for the
XT- plane, and the fixed line for the X-axis. For a point near the
origin, the shifts are
1) Memoire sur la torsion des prismes, avec des considerations sur leur
flexion. Memoires des Savants etr -angers, 1855.
Memoire sur la flexion des prismes. Journ. de Math, de Liouville, 2me Serie,
T. I, 1856.
2) That is, the stress on a plane parallel to a generator is only tangentij
and in the direction of the generator.
183] ST. VENANT'S PROBLEM. 479
^
dx
If the point is in the XY- plane, dz = 0, and as this plane is fixed,
w must be 0, necessitating
If the point is on the X-axis, dy = dz = 0, and as it must remain
on the X-axis, v and w must vanish, necessitating
(TT) = o-
\#SB/tl
The six conditions at the origin are accordingly,
while the conditions 67) everywhere satisfied are
72)
Of these the first two give
x du dv I, fciu dv 3w\ k / du 3w
fa===jy= ~2^\a^ + ^+ aij= ~2
^. ^w? dw
~ Z (I -\-ti~dz = ~ Vjz'
with the third
b) |^+|f = 0.
0^ r ^o;
The equations 144) of equilibrium are
73)
480 x- STATICS OF DEFORMABLE BODIES.
From the first of a) and b), by differentiation,
74) l**v gfj ^ 0**
From a),
n~\ Bu , Bv , dw ,+
The equations 73) become
c) ; ^
B*w
3*W . ^ V
and the conditions 71) at the origin, constitute the mathematical
statement of the problem.
Differentiating e) by #, and subtracting the derivative of c) by x
and of d) by y,
77)
Inserting the values of -«- and ^- in terms of -«- from a) gives
78) |^ = 0.
Adding the derivative of c) by y and of d) by x,
79)
2/ o
which, by use of b), gives
80) dxdydz = a
Differentiating e) by 0, and using 78),
= 0.
Differentiating c) by x, d) by y, and comparing with a),
82)
183] ST. TENANT'S PEOBLEM. 481
Comparing with 81), we find that both of these derivatives must be
zero. Accordingly we have
QON d2 (dw\ d2 fdw\ d2 (dw\ a2 /dw\ A
dF* \d~zj - df \R) = W (w = d^rty VW =
Hence •«- cannot contain any power of x, y or 0 above the first,
nor #?/, so that we may put
84) ^**<*
Hence from a),
r\ r\
85) Jx==Jy = ~
Differentiating 84), equations c) and d) become
Accordingly w and t; contain powers of 0 not above #3, M contains
powers of x not above ^2, v contains powers of y not above y2 and w,
powers of z not above #2.
Integrating 85) and 86),
1
87)
« =
-f
Putting these in b),
- ^ (
- ^ Ky + &i^) + ^y' W + /"' 0*0 = o,
so that ty, %j tpj f Sire of the second degree, in the variables indicated, sayr
Z(y) = a' -f < y + a2f t/2,
in which
< =
90)
vr~
, Dynamics.
482 X. STATICS OF DEFORMABLE BODIES.
Put
and we have
u = — rj lax + «j
91)
Integrating 84),
92) w = (a + atx + a,y) z + ^ (6 + \x + 6ay) + F(x, y),
which inserted in e) gives
If we put
94) F = & - (b ^±^-2 + \xf + b2x*y) -Vx- Wy
we get finally
or 5i is a harmonic function of the variables x, y. The conditions 71)
at the origin give
a1 = a"
We have now for the traction,
97) Z.-1. + *„£_*£
= j&fa + a^ 4- a^y + z(b + 6^ 4-
If we take for the origin the center of mass of the cross -section,
so that
C CxdS = C CydS = 0,
we have, integrating over the cross -section, z = const.,
98) C CzzdS = E C C(a -f Iz) dS =E(a + be) S.
183, 184J ST. TENANT'S PEOBLEM. 483
Now since the beam is in equilibrium, this integral Z- traction must
be the same for all values of £, hence & = 0. We have then finally,
[2-2 _ ^,2
ax + »i — g-^-
a202 b^z* , /8fl\
-v - ^ + « y 0- ^^
- y
From these we deduce the non- vanishing stress -components,
Z, = E |f = E [a + a
100)
y dw . du
and changing x to i/, alf as to &t, 62, j3 to — /3;
100) 2i-
The condition 69) at the cylindrical surface gives
-IAIN d& , s , da ' '":"
101) cosM + c
184. Determination of Function for Particular Cases.
Torsion. We may easily show that a function harmonic within a
certain plane contour, and on the contour satisfying the condition,
is uniquely determined. For if there be two functions £l± and i£2
satisfying the conditions, the difference V= fl± — *£2 is harmonic,
and on the contour
484 X. STATICS OF DEFORMABLE BODIES.
Now by Green's theorem, § 137, 55),
ff
=v ds ~7A Vdx dy'
and both integrals on the right vanish. Accordingly, as in the
demonstration of Dirichlet's Principle,
|^ = 1^ = 0, V = & - &2 = const
ex cy
But we have the condition &0 = 0, so that the difference V is zero,
and the solution is unique.
If the contour is a circle, the solution is immediate, by the aid
of circular harmonics. For the function & being developed in a
series of such,
103) &=(Anrnco$n(p -f B
, o
we have
104) ? = =nRn-\An cosncp
o
so that if f(Xj y) is given developed as a trigonometric series,
105) f(x, y) =(Cn cos ny + Dn sin + (l- *2 cos (ny) ,
184] TORSION OF PRISMS. 485
V, F15 F2 are uniquely determined, each vanishing at the origin. The
origin is any point on the line of centers of mass of the cross-
sections.
The most general solution of de Saint -Venant's problem now
contains six constants, a, a1? a2, /3, &1? &2. We may examine separately
the corresponding simple strains, by putting in each case all the
constants but one equal to zero.
I. a 4= 0.
u =
Z2 = Ea,
This is a stretch -squeeze of ratio ?y, which has been already treated
in § 176.
II. /3+0.
Ill) u = pyz, v = — pxs, w = pV.
These equations represent a torsion, whose rotation
is proportional to the distance from the origin, or fixed section.
We have u = v = w = 0 for x = y = 0 the line of centers. For
the stresses we find
112) Z.-E%-0, Zx =
so that the stress on any cross -section is completely tangential.
If the contour of the bar is circular, we have
cos (nx) : cos (ny) = x : y,
dV
so that ^n = 0 and V = tv = 0, and accordingly we see that the
plane cross -sections remain planes. For any other form of cross-
section than a circle, V does not vanish, so that the cross -sections
buckle into non- plane surfaces. This buckling was neglected in the
old Bernoulli -Euler theory of beams, and constitutes, as was shown
by de Saint -Venant, a serious defect in that theory, which is still
largely used by engineers.
In order to produce this torsion, we should apply at the end
cross -section the stresses
113) Zx=X, = iipy, Zy = Yz = -iipx,
that is at every point a tangential stress perpendicular to the radius
vector of the point in the plane and proportional to its distance
from the line of centers. As in practice it would be impossible to
486 X. STATICS OF DEFORMABLE BODIES.
apply stresses varying in this manner, we make use, in this and the
other cases, of de Saint -Tenant's principle of equipollent loads, viz:
If the cross- section is small with respect to the length, stresses
applied to a body near the ends produce approximately the same
effect if their statical resultants and moments are the same.
Consequently we may apply to the end faces or to the convex
surface near them the forces and couples,
X = I I X,dS = iifi I JydS = 0,
Y = / / F, dS = - [ift I i xdS = 0,
Z= I J Z2dS = 0,
114) L =(VZ> ~ *Y') dS =
M= C f(zXz- xZz}dS = pp C iyzdS = 0,
0 0
The twisting couple divided by the rotation per unit of length,
115) ^
(oz
is proportional to the fourth power of the radius. This is the law
announced by Coulomb, in his work on the torsion -balance. This
factor of the applied couple multiplying the twist per unit length is
called the torsional rigidity of the prism. Thus the shear -modulus p
may be determined by experiments on torsion.
For other contours than the circle it is convenient to introduce
into the problem instead of V its so-called conjugate function ^,
defined by the equations,
3V rOW cV dW
116) - = -jr-J H— = — -7f->
ex cy cy ex
184]
TORSION OF PRISMS.
487
If we now form the line - integral around the contour of the cross-
dV
section, of -? from 109) ,
on
dV
ll7) /ferfs=/(Ecos(^)+
= / [— y cos (nx) -\- x cos
7
ds
cos (nx) — — cos (nyy\ ds,
iayv'-vw"V $x
since if we circulate anti- clockwise (Fig. 152),
ds cos (w#) = dy, ds cos (ny) = — dx
the integrals become
118)
= - / (ydy + xdx),
*J
and since both differentials are perfect,
integrating,
119) gr^C-ite'+tf2).
Fig. 152.
Accordingly if we can determine a harmonic function *P which on
the contour shall have the value 119), the problem is solved. For
example take the functions,
giving
equal to a constant on the ellipse
tfz , y'
a2 -t- -ftl
if we take
giving
~ 2
488
X. STATICS OF DEFORMABLE BODIES.
We thus obtain
u**fl9*, v= -fxs, w
The curves of equal longitudinal
displacement, or contour-lines of
the buckled sections, are equi-
lateral hyperbolas (Fig. 153). The
stresses are, by 112),
Fig. 153.
The twisting couple is
f* f* f* r*
120) N= I I (xYz - yX,) dS = - ^t. / / (a*y* + tftf) dx dy>
In a similar we may deal with an equilateral triangular prism.
If a be the radius of the inscribed circle, the equation of the boundary
may be written,
(x -d)(x- i//3 + 2 a) (x + y V 3 + 2 a) = 0,
or
The function J.(.^3~3^2) is
a circular harmonic, and
is constant on the boundary if
we take
Fig. 154.
The curves of equal distortion
are shown in Fig. 154.
For any contour we have
for the couple, from 112)
121) N
(xY2 -
184, 185] TORSION OF PRISMS. 489
where I is the moment of inertia of the section about the
If we call
122) ff(X |T - y fl) dS = (1 - q)ff(*> + f) dS
then
123) N =
and the moment per unit twist per unit length qpl is called the
torsional rigidity of the bar, and q is de Saint -Tenant's torsion -factor.
For the circle q = 1, for the ellipse q = 2^fc2 2 , since I = ^-ab(a2+ &2).
If S = nab is the area of the ellipse, the rigidity may be written,
ii S*.
and by a generalization of this formula, de Saint -Venant writes
Having solved the problem for a great variety of sections, he found
that, when the section is not very elongated, and has no reentrant
angles, K varies only between .0228 and .026, its value for the ellipse
being .02533. We may thus put in practice
obtaining a most valuable engineering formula. Considering the
dimensions of S and I, we see that for similar cross -sections, the
rigidity varies as the square of the area of the section, as stated by
Coulomb, but for different sections the results differ much from those
of the old theory, in which q was supposed to be unity.
. 185. Flexion. For the third case we put
III. «,+(),
124) u = - y{£2-M02-r% v
125) Zz = Ed^ = Ealx)
The force on any section,
Z= I i ZzdS = Ea± I I
since the origin is the center of mass of the section. In this case
the couples L and N vanish, while
490 X. STATICS OF DEFORMABLE BODIES.
= I C(gX,-xZ,}dS= -Ec a parabola, or, since the displacement is supposed small,
a circle of radius — • This displacement is called uniform flexure, for
the curvature of the central line is constant. It is produced by the
action of no forces, but of a couple applied at the ends. The couple
is the same for all cross -sections, and is equal to the product of
Young's modulus by Iy and the curvature of the central line
127) M
This is the theorem of the bending moment. Such a strain is produced
in a bar when we take it in our hands and bend it by turning them
outwards. If the bar has a length I from the fixed section, the
deflection of the end is
128) M = _
and the flexural rigidity, or moment per unit displacement per unit
of length is 2EIy.
For a rectangular section of breadth 6 and height h,
I=
=J J
_A _A
¥ 2
For a circular section of radius R,
For a circular and rectangular beam of equal cross -sections, since
= bh the ratio of stiffnesses is
^rectangle _ bhs
and if & = 7&
Circle ' 12
•^circle
^ = 1.0472.
Since w = a^xz, a plane cross -section at a distance 0 from the
origin remains a plane which is rotated through the angle — = a^z,
185]
UNIFORM FLEXURE.
491
cutting the XY- plane at a distance - below the origin. Such a
plane remains normal to the line of centers, and the flexure is circular
(Fig. 155). If the section is rectangular, the sides y = + — become,
snce v = —
while the sides x = + -= become, since
or since g is constant, circles, so that the cross -section becomes like
Fig. 156.
. Fig. 155.
Pig. 156.
In an experiment by Cornu a bar of glass was thus bent, and
points of equal vertical displacement were observed by Newton's
fringes produced by a parallel plate of glass placed over the strained
plate. The curves u = const, are
#2 — ny^ = const.
a set of hyperbolas whose asymptotes make an angle a with the
Z-axis given by ctn2« = y. By photographing the fringes and
measuring the angle Cornu obtained a value of ^ very near to —>
the value given by Poisson.
The case a2 4= 0 *s precisely like that just treated, except that
the roles of the X and F-axes are interchanged. We pass therefore to:
492 X- STATICS OF DEFORMABLE BODIES.
IV. ^ 4= 0.
129) « = -^
«; = ^
where Fj is defined by
cos (w*) + (2 + ^) *y cos (»»)•
If the cross -section is symmetrical with respect to both axes
of X and Y, evidently the boundary condition is satisfied by a func-
3V
tion Fj_ evm in y, and o > — 1, we have
496
XI. HYDRODYNAMICS.
(2 + ,) a
which are satisfied by
We find the buckling of the sections
by 129),
Fig. 159.
In the case of a circular beam, we have
D = - — j and the curves of equal w are
the contour lines of which are shown in Fig. 159.
CHAPTER XL-
HYDRODYNAMICS.
186. Equations of Motion. The equations of hydrostatics 4)
§ 177 being
i)
where X, Y, Z are the components of the applied forces per unit
mass, we may obtain the equations of motion by d'Alembert's
Principle.
Suppose the velocity at any point in a perfect fluid of density $
is a vector q whose components u, v, w are uniform , continuous
differentiable functions of the point x, y, z and the time t. (The
notation is now changed from that of Chapter IX where u, Vj w denoted
displacements^) Then if we consider the motion of the fluid con-
tained in an element of volume dr of mass dm = $dt, we have the
effective forces
185, 186] EQUATIONS OF MOTION. 497
7 d*x -. du
gdrw = 9drjt,
o\ ^ d^u -, dv
2) (rdT-^^edr-^,
^ d*z 7 dw
Qdr^ = Qdr-d~,
and these are to be subtracted from the applied forces
and introduced in 1). Consequently we have the equations of motion
du\ dp
(x-
(z-
dw \ dp
~dt I ~" Js
Now by the ordinary derivative ^- is meant the rate of change
of velocity of a particular particle as it moves about. If we have
any function F pertaining to a particular particle we may write its
derivative
.^ dF = dF ^dFdx ,dF -j-i ~ are the velocity components of the
particle, u, v, w. Accordingly we have
KX dF 2F , dF , dF , dF
5) •^T~==^T + ^O h v Q h^^~-
dt dt dx ' dy dz
We shall call this mode of differentiation particle differentiation.1)
Introducing this terminology, dividing by Q and transposing, our
equations of motion 3) become
If +
du
U o p '
^^ . du
Vd-y + W8z
= X-
1 dp
o — '
Q dx
^\ dv .
6) 8t +
W ~ f- '
^iC '
v^- + w —
dy z
= r-
Q dy
dw
3«0 ,
div dw
#
1 dp
~dt +
^iC
dy dz
^Jz'
DF
1) In most English books the symbol •> used by Stokes, is used for
jj t
particle differentiation because of the very objectionable practice of making no
distinction in the symbol for ordinary and partial differentiation.
, Dynamics. 32
498 XT- HYDRODYNAMICS.
If we consider any closed surface fixed in space , expressing the
fact that the increase of mass of the fluid contained therein is
represented by the mass of fluid which flows into the surface, we
shall obtain an additional equation. The velocity being q} the volume
of fluid entering through an area dS in unit time is, as in § 169,
equation 78), qcos(q,n)dS, and the mass, pgcos (q, n) dS. We
have therefore for the total amount entering in unit time
7) I / gq cos (q,ri)dS =1 I p{iecos(w#)-f- vcos(ny)-\-wcos(n0)}dS
But this is equal to the increase of mass per unit of time,
for the volume of integration is fixed, that is, independent of the
time, consequently we may differentiate under the integral sign.
Writing this equal to the volume integral in 7) and transposing,
Since this holds true for any volume whatever the integrand must
vanish, so that we have
_ A
^t~ ~^r + ~W~ ~0^~
which is known as the Equation of Continuity.
Performing the differentiations we have
00 , 00 , 00 , 00 , (0** , 0^ .
or in the notation of particle differentiation,
If now we fix our attention upon a small portion of the fluid
of volume V as it moves, its mass will be constant, say, m = $ V= const.
By logarithmic differentiation,
I^ + ^-^-O
g dt H F dt ~
so that the expression
... du.dv . div _ 1 dQ _ 1 dV
^^J^^ Tz~ ~^~di~=T^dt'
that is, the divergence of the velocity is the time rate of increase
of volume per unit volume. This corresponds with the expression
186, 187] EQUATION OF CONTINUITY. 499
for the dilatation found in equation 75) § 169, the divergence, which
we shall still call , being now the time rate of dilatation. Accord-
ingly the equation of continuity is purely kinematical in character
and expresses the conservation of mass of every part of the .fluid.
If the fluid is incompressible, Q is constant, and consequently
that is, the velocity of an incompressible fluid is a solenoidal vector.
This is the property that give the name to such vectors, and we
see, as in § 117, that the flux across every cross -section of a tube
of flow is constant.
Besides the three dynamical and one kinematical equation there
will be a physical equation involving the nature of the fluid, giving
the relation connecting the density with the pressure,-
making five equations to determine the five functions u,v,
of the four variables x^y^z^t.
We have here made use of two distinct methods. In one we
fix our attention on a definite point in space and consider what
takes place there as diiferent particles of fluid pass through it. This
is called the statistical method, for by the statistics of all points we
get a complete statement of the motion. This method is commonly
associated with the name of Euler and the equations 6) are called
the Eulerian equations of motion. The second method consists in
fixing our attention upon a given particle and following it in its
travels. In this we use the notation of ordinary derivatives. This
is called the historical or Lagrangian method. Obviously if we know
the history of all particles we also have a complete representation
of the motion. Both methods are due to Euler. We shall not here
make use of the Lagrangian equations and shall therefore not write
them down. The student will find them in the usual treatises on
Hydrodynamics of which Lamb's and Basset's Hydrodynamics, Kirch-
hoffs Dynamik and Wien's Hydrodynamik may be especially com-
mended.
187. Hamilton's Principle. We shall now deduce the equa-
tions by means of Hamilton's Principle.
The kinetic energy of the fluid contained in an element dr
being -~gdv times the square of its velocity, we have for the kinetic
energy of the fluid contained in a given fixed volume,
32*
500 XI- HYDRODYNAMICS.
For the potential energy, besides that due to the external forces,
for which
17) dW = - {Xdx + Tdy + Z8g}dv,
we have the energy stored up by the pressure doing work in com-
pressing the fluid, if it be not incompressible. This potential energy
is the dW of equation 122) § 172. If we put for the displacements
of the particle dx, dy, dz, we have
j.
os* = -K— ; os,, = -~-i
ox dy
Putting
Px = Py = Pz = -p,
Tx = Ty = T2 = 0,
we have for this part of the energy, sometimes called intrinsic energy,
We accordingly have as the equations of Hamilton's Principle,
C(dT - dW' - dW") dt = 0,
Performing the variations,
20) fdt\ (f{\9&t dt , ,
Integrating the first three terms by parts with respect to t
we have,
187, 188] HAMILTON'S PRINCIPLE. 501
*l
J
-dt 9 Sx + Sy + | i, - (Xix
The integrated terms vanish as usual at the limits for the time. We
may now integrate the last three terms with respect to the space
variables , obtaining
= / IP (®x cos (nx) + $y cos (ny) + d0 cos
fff &.'*+%«+
+
If we assume that dx, dy, d# vanish for the particles of the
fluid at the bounding surface, the surface integral vanishes. We
therefore have, collecting the terms according to 8x, dy, dz,
23)
By the usual reasoning the coefficients of dx, 8y, dz must vanish,
giving us the equations of motion 6).
188. Equation of Activity. Subtracting from both sides of
the first equation 6) the quantity
9N du . dv . dw
we obtain X1
3 / 2
"
^ du , /du dw\ /dv du\
25) ^T + ^(^ — o- — v(o -- TT-
' ^^ ' w dx) \dx dj
TT-
dy
If the applied forces are conservative and derived from a potential F,
the right-hand member is the derivative,
502 XL HYDRODYNAMICS.
where P has the value of § 177, 5). In the left-hand member occur
in this and the companion equations the expressions
i /£. _ *V >
'2 \aa; ds// ~ S,
which represent the components of a vector co, which by comparison
with the expressions for &x, &y, &z in § 169, 77) is seen to be the
angular velocity of rotation of a particle or the vorticity. Let us
accordingly write, putting
v, ^w; and adding, the terms in
1, 17, £ disappear and integrating over any volume, we have
- / / [« cos (wa;) + » cos (ny) + «c cos (»«)] ( UQ —p) dS
^ -L ?^") -L
~ '
Introducing the value of U and of its multiplier in the last integrand,
o
-ST*
29)
o
which by equation 10) is equal to - • -ST* an^ transposing, we obtain,
If the applied forces are independent of the time, -^r = 0, and we
may write the left-hand member,
188, 189] EQUATION OF ACTIVITY. 503
30)
since the volume of integration is fixed. The first term of the integral
represents the kinetic energy and the second term, the potential
energy due to the applied forces. The term on the left in 29) is
accordingly the rate of increase of the energy, kinetic and potential.
Of the terms on the right, the amount of matter Qqcos(qn)dS
flowing through dS in unit time brings with it the energy
so that the first part of the surface integral represents the total in-
flow of energy. The remaining surface integral and volume integral
containing p represent the work done by the pressure, for at the
surface the velocity q and the force pdS give the activity
pqcos (qri)dS,
so that the surface integral represents the activity of the pressure at
the surface.
If we consider a small element of volume F, the work done in
compressing it by an amount dV is as above —pdV, and the activity
Q1\ dV du . dv
31) -r- =
Putting V=dr and integrating, we find that
rrr isu . a.
is the activity of the pressure in producing changes of density in
the whole mass. Transposing this term we find that equation 29)
expresses the following: The rate of increase of energy of the fluid,
both kinetic and potential, due to the external forces plus the activity
of compression (production of intrinsic energy) is equal to the rate
of inflow of energy plus the activity of pressure at the surface.
Equation 29) is therefore the equation of activity or conservation of
energy.
189. Steady Motion. Steady motion is defined as a motion
which is the same at all times. Assuming that not only X, T9 Z, V
but^f, v,w,p, Q are independent of t, equations 27) for steady motion
become
32)
504
XL HYDRODYNAMICS.
If the motion be non- vortical, the left-hand members vanish,
and we immediately obtain the integral
33)
F + P -f 4 = const,
for the expression on the left has been assumed independent of t,
and by the equations is shown to be independent of x, y, z.
It is to be noticed that if we multiply equations 32) respectively
by u, v, wy or by |, 77, £ and add, the left-hand member vanishes
identically. But the operator u^ — h^o h w ~— denotes differentiation
in the direction of the line of the vector- velocity q, or stream-line
r\ r\ r\
(see p. 333), and | ~ — l~ ^2 — ^ ^7T differentiation in the direction
of the line of the vector co, or vortex -line. Consequently even though
there is vortical motion, along a stream -line or a vortex -line the
sum F -f P -f Y *f *s constant in steady motion, though its value
changes as we go from one line to another.
If the fluid is incompressible P = — , if there are no applied
forces F=0, and equation 33) becomes
34)
P 1
=~ = const. — —
so that where the velocity is small the pressure
is great and vice versa. By constricting a tube
the velocity is made large and the pressure
accordingly is smaller than at other parts of the
tube. This is the principle of jet exhaust pumps,
like that of Bunsen (Fig. 160), the air being
sucked in at the narrow portion of the jet. The
same principle is made use of in the Venturi
water-meter. The main being reduced in diameter
at a certain portion and the difference of pressure
at that point and in the main being measured,
the velocity is computed. If the pressure at two
cross -sections S1 and S2 are p± and p2 we have
35)
Fig. 160.
or
36)
But by the equation of continuity, the velocity being solenoidal,
37)
189] STEADY MOTION. 505
Combining this with equation 36),
38) #i-i>a = -S0
which determines q± in terms of the difference of pressures. The
flux in unit time is then
The theorem expressed by equation 33) is known as Daniel Bernoulli's
theorem.
For gases expanding isothermally
P = a log p = a logp -f const.
Consequently equation 33) becomes
39) a logp -}- -— q2 = const.
This formula may be used to calculate the velocity of efflux through
an orifice from a vessel containing gas under pressure. If the pressure
in the vessel at a point so remote from the orifice that the air may
be considered at rest is p and if the pressure of the atmosphere at
the orifice where the velocity is q is p0, we have
alogp = alogpQ + y£2,
\^
40) (f = 2 a log — •
If the efflux is adiabatic, as in practice it nearly is, by § 178, 18)
Accordingly
41) q2 = \
which is the usual formula for the efflux of gases.
If the external force is gravity V=g0, so that equation 33)
becomes for an incompressible fluid,
42) — -f qz + -^ q2 = const.
Q 2
If we consider efflux from a reservoir whose upper free surface is
so large that q is negligible, the pressure being that of the atmo-
506 XI. HYDRODYNAMICS.
sphere, the £- coordinate zl9 the velocity of efflux q at a point where
z = #2 ig given "by
43) a'
or the velocity of efflux is equal to that acquired by a body falling
freely from a height equal to that of the free surface to the orifice.
This is Torricelli's theorem.
190. Circulation. We define the circulation along any path
as the line integral of the resolved tangential velocity,
B B
44) v = -^-9 w = -£->
ox £y dz
and qp is called the velocity potential, a term introduced by Lagrange.
(When there is vorticity there is no velocity potential.)
Before 1858 only cases of motion had been treated in which a
velocity potential existed. In that year appeared the remarkable paper
by Helmholtz 1) on Yortex Motion.
Let us now find the change of circulation along a path moving
with the fluid, that is, composed
of the same particles, the forces
being conservative.
Our equations of motion 3) may
be written, putting U' = — (F-f P)
49)
du
~dt
dv
dw
~dt
dU'
vdz
dU'
Fig. 161.
The change of circulation along the path AB is
50)
dt
d
dt
A
B
I (udx -f vdy + wdz),
in which dx9 dy, dz vary with the time, being the projections of an
arc ds composed of parts which move about. If after a time dt the
arc ds assumes a length ds' whose components are dx', dy', dz* we
have (Fig. 161)
1) Vber Integrate der hydrodynamischen Gleichungen, welche den Wirbel-
bewegungen entsprechen. Wissenschaftliche Abhandkmgen I, p. 101.
508 XL HYDRODYNAMICS.
' = x + dx + dt [u + <^-dx + ^dy + —
- (x + udf)
cy z
and therefore the change per unit time in the projections are
du 7
K.I\ d /? \ dv -, . dv ^ , cv
51)
Thus we have
B
KON ^w , dv dw
52)
and substituting from equations 49) and 51)
B
-ON dffAB C[3U' . cU' . 8U'
53) -dr=Jb^dx + i^dy + -wd*
A
du . 3v . 3w
^ + Vd-x + WZx
du . 3v , dw
^ + v8j + w
du . dv .
A
which vanishes for a closed curve.
Therefore if the forces are conservative, the circulation around
any closed path moving with the fluid is independent of the time.
Thus if the circulation around any closed path is zero at one time,
it is always zero, or in other words if a velocity potential once
exists, it always exists. This theorem is due to Lagrange.
190, 191] VORTEX MOTION. 509
191. Vortex Motion. We will now consider the case in which
no velocity potential exists , that is, the case of vortex -motion,
according to the methods of Helmholtz.
From the equations 27), whose right-hand members are the
derivatives of — ( V + P + ^ en , this quantity may be eliminated by
differentiation. Differentiating the last equation by y, the second by #,
and subtracting, we obtain
*Wis U dz ~*~ * dz W dz '
or otherwise
-•I!-:
On the right the coefficient of w vanishes identically by 47), and
that of | is by the e
equation 55) becomes
that of | is by the equation of continuity 12) equal to - ~> thus
Now we have
£ M ^ *| _ I *i
® dt\Q/ dt Q dt
and accordingly we may write our equation 56) and its two companions
i ^ , 1 ^ . 1 ^
9 d# 9 ay 9 2z'
rrrx _ | ^V 7] ^V ^V
^i-Si"^-??? • e 27'
I fiw r\ diu g Bw
7 ^ 7 3y "^7 a« '
Thus the time derivatives of — > — > — for a given particle are homo-
geneous linear functions of these quantities. By continued differentiation
with respect to t and substitution of the derivatives from these equa-
tions, we see that all the time derivatives are homogeneous linear
functions of the three quantities themselves. Consequently if at a
certain instant a particle does not rotate, it never acquires a rotation.
This we find by developing —t—>— as functions of t by Taylor's
theorem, for if the derivatives of every order vanish for a certain
510
XL HYDRODYNAMICS.
instant, the function always vanishes. Stokes1) objects to this method
of proof as not rigorous inasmuch as it is not evident that the func-
tions I, rjf £ can be developed by Taylor's theorem, and replaces it
by the following demonstration.
Let L be a superior limit to the numerical values of the coef-
ficients of — 9 — i — in the second member of equations 57). Then
Q Q Q
evidently £, 17, £ cannot increase faster than if their numerical or
absolute values satisfied the equations
58)
A
dt
_d
dt
d
dt \ Q
instead of 57), |, ??, g vanishing in this case also when # =
these three equations and writing
Adding
we obtain
59) ^
The integral of this equation is
and since 52 = 0 when t = 0, c must be zero, and £1 is always zero.
Since the sum of the absolute values cannot vanish unless the separate
values vanish, the theorem is proved.
Let us now consider two points A and B lying on the same
vortex line at a distance apart ds = s — > where s is a small constant.
Since the particles lie on a vortex -line we have
£xvx dx dy dz ds e
We have for the difference of velocity at A and B
£1\ ^U J %U 7 VU J
bl ) Un — MA == o — u>% ~~r ^r~ dy ~r ~^~ u>%
or by equations 57),
62)
J_
Q dx^ Q
1) Stokes, Math, and Phys. Papers, Vol. II, p. 36.
191, 192] VORTEX -MOTION. 511
Now at an instant later by dt, when the particles are at A' and Bf,
we have
dx' = dx + (UB - O dt = 8 +
63) dy' = dy + (u* - UA} dt = sQ+± g) dt\
de' = dz + (UB - UA) dt = e[± + ± g ) dt]
Therefore the projections of the arc ds' in the new position are
proportional to the new values of — > — ? — > as they originally were,
so that the particles still lie on a vortex -line. Accordingly a vortex-
line is always composed of the same particles of fluid. Also since
the components of ds have increased or have changed so as to be
always proportional to the components of — ? if the liquid is in-
compressible the rotation is proportional to the distance between the
particles. And whether Q vary or not, if S be the area of the cross-
section of a vortex -filament, gSds, the mass of a length ds remaining
constant, so does So, the strength of the filament.
It is easy to see that this is equivalent to a statement of the
conservation of angular momentum for each portion of the fluid.
Evidently in a perfect fluid no moment can be exerted on any portion,
since the tangential forces vanish.
Accordingly the strength of a vortex -filament is constant, not
only at all points in the filament but at all times, consequently a
vortex existing in a perfect fluid is indestructible, however it may
move. It is from this remarkable property of vortices discovered
by Helmholtz that Lord Kelvin was lead to imagine atoms as COD-
sisting of vortices in a perfect fluid.
192. Vector Potential. Helmholtz's Theorem. We have
seen that any curl is a solenoidal vector. We may naturally ask
whether conversely any solenoidal vector can be replaced by the curl
of another vector. It was shown by Helmholtz that any uniform
continuous vector point -function vanishing at infinity can be expressed
as the sum of a lamellar and a solenoidal part, and the solenoidal
part may be expressed as the curl of a vector point-function. A
vector point -function is completely determined if its divergence and
curl are everywhere given. Let q be the given vector, which in our
case is the velocity of the fluid. Let us suppose that it is possible
to express it as the sum of the vector -parameter of a scalar func-
512 XI. HYDRODYNAMICS.
tion (p and the curl of a vector function Q, whose components are
U, V, W. Accordingly let us put
_
-
64)
dt dx dy
Finding first the divergence of q we have
f>*\ j- du . dv . dw
bo) div. q — % ho "~ -Q— =
the divergence of the curl part vanishing. But by § 128, 5) we
know that if cp and its first derivatives are everywhere finite and
continuous, we have
ST. a 7
— dr.
Since q is continuous by hypothesis, div. q is finite. Consequently the
lamellar part of q is determined by its divergence.
Secondly finding the curl of q,
dy
All 4- dU 4- dV 4- dW
-^U + + +
1Z_ zu\ _ —^ > — respectively, is called the vector potential
of the vector — -• We may thus abbreviate our results in the vector
_ Tt
equations,
71) q = vector parameter (u* + v* + w^ dr
dlJ cW
V\
- - Ti)
= 2 / / K*7 w - w F) cos (WflO + (wU—uW) cos (ny)
-f
If the integral be taken over all space, since the motion is supposed
to vanish at infinity the surface integrals vanish, and
81) T= <>fff[ Vt + Vrt + Wf\ dr,
or inserting the values of U, V, W from 70)
and the integration may now be restricted to the vortices.
If again we integrate by filaments, we find
where the integration is expressed as over the length of each of the
double infinity of vortex -filaments constituting the vortices. This is
the form obtained for the energy of two electric currents by Franz
Neumann.
195. Straight parallel Vortices. Let us now consider the
case in which the vorticity is everywhere parallel to a single direc-
tion, that of the axis of g. Let the motion be uniplanar, that is
parallel to a single plane, the X3^-plane, and the same in all planes
parallel to it. All quantities are therefore independent of g. The
vortices are columnar and either of infinite length or end at the free
surface of the liquid. Such vortices may be produced standing
vertically in a tank with a horizontal bottom. Under the conditions
imposed we have
QA\ du dv CM c. rr rr
84) 0 = w = TT- = Q- = ^ = $ = 7? = c7=F,
dz dz dz
and
QKX dW dW
85) u = -5—> v = -- ;—>
cy Ox
33*
516 XI. HYDRODYNAMICS.
86) 2? = ^-|'
and by equations 57)
so that £ is independent of the time for any given vortex -filament.
The function W is not a velocity potential, but is said to be
conjugate to a velocity potential (p for which
87) — fe — If-
The function TF has a simple physical meaning. If we find the
amount of liquid which flows across a cylindrical surface with
generators parallel to the #-axis of height unity in unit of time,
we have B B
. 88) iff = I q cos (qn) ds = I [u cos (nx) + v cos (ny)\ ds,
A A
the line integral being taken around any orthogonal section of the
cylinder. Now we have
ds cos (nx) = dy,
ds cos (ny) = — dx,
so that
89) * •-{ ~ vdx) -dx + dy -W,- WA.
A function, the difference of whose values at two points A and J5
gives the quantity flowing in unit time across a cylinder of unit
height drawn on any curve with ends at A and jE?, is called a flux
or current function. The quantity crossing is independent of the
curve because the fluid is incompressible. In the present case the
vector potential "FT is a current function. The stream lines being
lines across which no current flows are given by the equation ^ = const.
Substituting the values of u and v from 85) in 86), we have
But this is the equation for a logarithmic potential with density —
§ 138, 61), so that we have as the integral
91) W=-
as may also be found from equation 70) by integrating over the
infinite cylinder as in § 135, subject to the difficulty mentioned on
p. 385. The value of W given in equation 91) satisfies the equation
195] COLUMNAR VORTICES. 517
outside of the vortices and equation 90) at points within them, as
shown in § 138. If we have a single vortex filament of cross-section
dS and strength n = £'dS,
93) W = - ^-logrdS=- -logr,
7T 7C
and the lines of flow are circles, r = const. Then
= =^ y-y
dy 'XT' r
cW x x-x1
V =
ex Ttr r
95)
,2
the velocity is perpendicular to the radius joining the point x, y
with the vortex and inversely proportional to its length. It is to be
observed that although the motion is whirling, every point describing
a circle about the center, the motion is irrotational except at the
center, where the vortex -filament is situated, each particle describing
its path without turning about itself, like a body of soldiers obliquing
or changing direction while each man faces in the same unchanging
direction. The motion in the vortex on the contrary is similar to
that of a body of soldiers wheeling or changing direction like a rigid
body rotating.
If we have a number of vortices of strengths ^, 3«2, . . . %„, and
form the linear functions of the velocities of each,
Z7
r,
where u,9 vsy is the velocity at the vortex s both vanish. For any
pair of vortices r and s we have
where ur is the part of the velocity at xr, yr due to the vortex of
strength x, situated at xsys. Thus
while similarly
xr-xs
XSUS = KsXr -^— »—>
so that the terms of the sum destroy each other in pairs.
1) U and V have nothing here to do with the components of the vector-
potential.
518 XL HYDRODYNAMICS.
Similarly for vortices continuously distributed, the strength of
any elementary filament being
97) U
which again vanishes, since every point is covered by both dS and dS'.
If we define the center of the vortex as x where
then since g does not depend upon t, if we follow the particle
differentiating,
99)
the integrals being taken over areas moving with the liquid. Therefore
or the center of all columnar vortices present remains at rest.
If we have a single vortex filament of infinitesimal cross- section S,
for which
101)
the velocity depends on the current function W = - — log r. In the
vortex and close to it, if x is finite, g, W, u, v are infinite. But at
the center u = v = 0, the vortex stands still and the fluid moves about
it in circles with velocity — The angular velocity and the area of
the cross -section remain constant, although the shape of the latter
may vary. If we have two such vortex-
filaments each urges the other in a direc-
tion perpendicular to the line joining
them, they accordingly revolve about their
center, maintaining a constant distance
from each other. If they are whirling in
the same direction the center is between
them (Fig. 162), but if in opposite direc-
tions, it is outside, and if they are equal
it lies at infinity. Such a pair of vortices
may be called a vortex-couple or doublet,
jijg 162 and they advance at a constant velocity,
keeping symmetrical with respect to the
plane bisecting perpendicularly the line joining them. This plane is
a stream -plane and may accordingly be taken as a boundary of the
195] VORTEX PAIRS. 519
fluid. Since either vortex moves with a velocity - - and half way
between them the velocity being due to both is - — = —^ we find
~¥~
that a single vortex near a plane wall moves parallel to it with a
velocity one fourth that of the water at the wall. This is an
illustration of the method of
images, of frequent application
in hydrodynamics.
As another illustration con-
sider the motion of a single
vortex-filament in a square corner
inclosed by two infinite walls.
The motion is evidently the same
as if we had a pair of vortex-
couples formed by the given Q A O
vortex and its images in the
two walls, turning as shown in
Fig. 163 and forming what may p. 16g
be called a vortex kaleidoscope.
From the symmetry it is evident that the planes of the walls are
stream -planes, so that we may consider the motion in one corner
alone. If x and y be the coordinates of the vortex considered, we
have as due to the others,
Y. K y Y. x*
Zn x*-\-y* 2 it y (x2 -f
102)
xx x Y, y*
' ' *~* = ~ *
Since u and v are the velocities -jr>. -~ of the vortex, we have for
at at
the equation of its path
dx
irvoN dt u xs dx
103) -j— = — = -- - = —, or
ay v y ay
dt
dx _ dy
!c*~ ~~y*'
whose integral is
111 T12?/2
104) + = ' ** + f = -'
= -, sin2 & cos2 ^,
n" '
a
and in polar coordinates,
105)
the equation of a Cotes's spiral, having one of the axes as an asymptote.
520 XI. HYDRODYNAMICS.
The same problem gives us the motion of two equal vortex-
couples approaching each other head on, or a single vortex- couple
approaching a plane boundary, showing how as they are stopped they
spread out. The beharior of vortex -couples will serve to illustrate
that of circular vortex rings, for the theory of which the reader is
referred to Helmholtz's original paper. The two opposite parts of a
circular vortex appear to be rotating in opposite directions if viewed
on their intersection by a diametral plane normal to the circle, thus
resembling a vortex -couple. It is found that the circular vortex
advances with a constant velocity in the direction of the fluid in the
center, maintaining its diameter, but that when approaching a wall
head on it spreads out like the vortex -couple. Two circular vortices
approaching each other do the same thing, but if moving in the
same direction the forward one spreads out, the following one
contracts and is sucked through the foremost vortex, when it in turn
spreads out and the one which is now behind passes through it, and
so on in turn, as may also be shown for two columnar vortex-couples
traveling in the same direction.
Most of these properties of circular vortices may be realized
with smoke rings made by causing smoke to puff out through a
circular hole in a box, or mouth of a smoker, or smoke-stack of a
locomotive. The friction at the edge of the hole holds the outside
of the smoke back, while the inside goes forward, establishing thereby
the vortical rotation. As previously stated no vortex could be formed
if there were no friction. It is to be noticed that the direction of
the fluid on the inside of the vortex gives the direction of advance.
196. Irrotational Motion. We shall now consider the non-
vortical motion of an incompressible fluid. We then have a velocity
potential qp and
-t r\r>\ d
v = ^-i w = ^--
ox oy dz
The equation of continuity becomes
107) 4y = 0,
and the potential is harmonic at all points except where liquid is
being created (sources) or withdrawn (sinks). The volume of flow
per unit time outward from any closed surface S is
108) - / / [u cos (nx) -f v cos (ny) -f- w cos (nzj] dS
so that if this is not equal to zero, it is equal to the quantity created
in the space considered in unit time,
195, 196, 197] IRROTATIONAL MOTION. 521
so that if we put z/qp = tf , 6 is the amount of liquid produced per
unit volume per unit of time. The total amount
dt — i i i Gdr
is called the strength of the source. If (3 is given as a function of the
point we have
Accordingly the velocity potential has the properties of a force
potential, the density of attracting matter being represented by
L times the strength of source per unit volume. The negative sign
occurs here from the different convention employed, it being customary
to define the force as the negative parameter, the velocity as the
positive parameter of its potential. • In particular a point source of
strength m produces a radial velocity of magnitude -^—^ This
system is called by Clifford a squirt.
197. Uniplanar Motion. A simple and interesting case is that
of uniplanar flow as defined above. We then have all quantities
independent of s, so that Laplace's equation reduces to
nn
A powerful method of treatment of such problems is furnished by
the method of functions of a complex variable. The complex number
a 4- ib, where a and & are real numbers and i is a unit defined by
the equation
»»=-i,
(the same root being always taken) is subject to all the laws of
algebra, and vanishes only when a and ~b both vanish separately.
Any function of the complex number obtained by algebraic operations,
after substituting for every factor i2 its value — 1, becomes the sum
of a real number plus a pure imaginary, that is a real number
multiplied by i. Any equation between complex numbers is equivalent
to two equations between real numbers, being satisfied only when
the real parts in both numbers are equal as well as the real coeffi-
cients of i in both members. If z denote the complex variable x + iy,
any function of & may be written
w = f(z) ==• u + iVj
522 XI. HYDRODYNAMICS.
where u and v are real functions of the two real variables x and y.
For instance
z* = C» + ty)8 = x* - i
w = #2 — ?/2, v
1 _ 1 x — iy
z
U =
Let us examine the relation between an infinitesimal change in z
and the corresponding change in /"(#)• We have, x and y being real
variables capable of independent variation,
114) dz = dx + i dy,
115)
Consequently by division, •
du , d«* . /#« , dv \
7 i . -, o — dx •+• -^— dy -+- 1 1 75 — dx -+- -^ — a w I
^.,,N die du -\-idv dx dy \o x cy }
dyl dx
The ratio of the differentials of w and z accordingly depends in
general on the ratio of dy to dx, that is, if x and y represent the
coordinates of a point in a plane, on the direction of leaving the
point. If the ratio of dw to dz is to be independent of this direc-
tion and to depend only on the position of the point x, y, the
numerator must be a multiple of the denominator, so that the expression
containing ~ divides out. In order that this may be true we must have
du . . dv
that is
117)
\dx n dx/ dy dy
Putting real and imaginary parts on both sides equal,
-,^Q^ du cv dv du
HO) O— = ^~ > 7T~ = a" >
dx cy ex cy
and
11Q. dw du . dv dv .du
= + * = -'
197] COMPLEX VARIABLE. 523
In this case the function w is said to have a definite derivative defined by
j,f , N j. dw
dy = 0
and it is only when the functions u and v satisfy these conditions 118)
that u + iv is said to be an analytic function of g. This is Riemann's
definition of a function of a complex variable.1) The real functions
u and v are said to be conjugate functions of the real variables x, y.
It is obvious that if w is given as an analytic expression
involving 0, w = f(&), then w always satisfies this condition. For
dw df(z] dz f'C\ dw df(z] dz . ~, / x
dx dz dx I \ )i fly d% fly I \ )'
Accordingly
.dw . (du . dv\ dw cu . dv
cu _ dv dv _ du
dx dy' ex dy
If we differentiate the equations 118), the first by x, and the second
by y and add, since
d*v dzv
dxdy ~ dydx'
we obtain
120^ — 4-— — 0
dx*^ dy*-"'
Differentiating the second by x and the first by y and subtracting,
we find that v satisfies the same equation
Thus every function of a complex variable gives a pair of solutions
of Laplace's equation, either one of which may be taken for the
velocity potential, representing two different states of flow.
It is to be noticed that the question here dealt with is simply
one of kinematics, since Laplace's equation is simply the equation of
continuity and there is no reference to the dynamical equations.
The question arises whether any two solutions of Laplace's
equation will conversely give us the function of a complex variable.
It obviously will not answer to take any two harmonic functions,
for they must be related so as to satisfy the equations 118) or be
mutually conjugate. In order to avoid confusion with the velocity
1) Riemann, Mathematische Werke, p. 5.
524 XL HYDRODYNAMICS.
components u and v, let us call the two conjugate functions (p and ^7
satisfying the equations
It is evident that qp and ^ have the relation of the velocity potential
and stream function denned in § 195. If one function is given we
can find the conjugate, for we must have
which by equations 122) is
Now if we call this Xdx + Ydy it satisfies the condition for a
perfect differential
dX __ dY
dy dx
that is, in this case,
dy* dx*
Consequently the line integral
dtp -. . dm -. }
— dx + 7T2- dy \
dy dx y)
from a given point #0, yQ to a variable point x, y, is a function only
of its upper limit and represents ^. Similarly if ^ is given
123) ,
Furthermore the first of the equations 122) is the condition that
V dx + ? ^F which in virtue of the equations
d$ dW
ib = 75— = -- o— 9
dx dy
125) = a* = ^
^ 3y dx'
are conjugate to each other and give a new analytic function of z,
197]
CONJUGATE FUNCTIONS.
525
whose derivative is gp + iif>. From these by new integrations we
may obtain any number. The method of the complex variable accord-
ingly gives us the solution of an unlimited number of uniplanar
problems.
The equations 122) are geometrically the condition that the lines
(p = const., iff = const, intersect each other everywhere at right angles.
If ijj is the stream function the lines ijj = const, are the lines of flow,
which we know intersect the equipotential surfaces at right angles.
As examples consider the cases worked above,
w = £2, (p = x2 — 2/2, $ = %xy.
The equipotential lines are sets of equilateral hyperbolas, intersected
at right angles by the system of equilateral hyperbolas forming the
stream lines (Fig. 164). The stream line iff = 0 consists of the X and
Y axes, which may accordingly be a boundary, so that one quarter
Fig. 164.
of the figure represents the flow in a square corner of a stream of
infinite extent.
The function w = — gives
x .1. _ y
526
XL HYDRODYNAMICS.
The equipotential lines give a set of circles all tangent to the Y-axis
at the origin , while the lines of flow are a similar set all tangent
to the X-axis (Fig. 165). The water flows in on one side of the
Fig. 165.
origin and out at the other as if there were a source on one side
and an equal sink on the other close together.
The function zn, of which the two examples just treated are
particular cases, gives an interesting case which is most simply worked
out by the introduction of polar coordinates.
x = r cos G), y = r sin o>,
z = x + iy = r (cos co + i sin CD) = rei(a,
-f
from which we obtain the two conjugate functions
126) u =
If we multiply these two harmonic functions by constants and add,
the sum
127)
rn [An cos (wo) + -#« sin n&
is the circular harmonic function treated in § 140. We may accord-
ingly develop the velocity potential in a series of circular harmonics,
197] FLOW AROUND CORNER. 527
128) y
and if we know the values of qp on the circumference of a circle
with center at the origin, we may find the coefficients by the method
of Fourier as in § 140 a, 83).
Let us examine the motion in a segment between two walls
making an angle 2 a at the origin and reaching to infinity. If we
use the value of cp given by equation 128), the coefficients and the
values of n admissible are to be determined by the condition
along each wall. But since dn = rda), we have
129) - ^
which must vanish for to = + a. If
a*-f-i
na = — ~ — n,
and if
ncc = nit, sin (+ no) — 0,
3c being any integer. Therefore if we put when n is an odd multiple
of — 9 An = 0 and Bn = C^x+i and for even multiples, Bn = 0 and
An = C^x, we shall have as a solution of the problem
n 2 a • /2x-fl n V-.--AT « /»* •
C2y. + ir sm (-— ^— • — '») + ft^ cos ^— o
The tangential velocity at the wall is given by
131)
The exponent of the lowest power of r is ^— -1. If this is negative,
that is if a > --> the velocity is infinite for r = 0, that is at the
corner, unless C^ = 0.
The pressure is given by the equation
p = const. — -| £2,
so that at a sharp projecting edge around which the water flows
there would be an infinite negative pressure. This being impossible,
around such an edge the motion is discontinuous, so that instead of
528
XL HYDRODYNAMICS.
flowing as in a) Fig. 166, the water flows as in b), the flow being
discontinuous at the dotted line. In actual fluids such surfaces of
discontinuity give rise to vortex motion, so that we see eddies formed
at projecting corners.
a)
The function
Fig. 166.
with
= logr 4-
gives us radial stream lines forming a uniplanar squirt, while
y> = co, ijf = log r
Fig. 167.
and for a free surface, p = 0,
133) C -
gives us flow
in circles with
a velocity
dcp
The velocity at the center is
infinite. This flow is exactly
what we found in § 195 to be
produced by a vortex filament
at the center.
If the fluid is under the
influence of gravity, we have
42) p = const. — gqz — —
197, 198] WAVE MOTION. 529
If z is zero when r = , 0 = 0, and the equation of the sur-
face is
The form of the surface is shown in Fig. 167. This is approximately
the form taken by the water running out of a circular orifice in the
bottom of a tank, although the above investigation takes no account
of the vertical motion.
198. Wave Motion. The case of uniplanar water waves may
be dealt with by the method of the preceding section.1) Let us take
the XY- plane vertical, the Y-axis pointing vertically upward and
the motion as before independent of the 0 coordinate, so that we
may use e to denote the complex variable. We shall find that the
waves travel with a constant velocity and iib will therefore simplify
the problem if we impress upon the whole mass of liquid an equal
and opposite velocity so that the waves stand still and the motion
is steady. Such still waves are actually seen on the surface of a
running stream.
Let us first consider waves in very deep water. At a great depth
the vertical motion will disappear and we shall have only the constant
horizontal velocity that we have impressed, so that
u = — a, v = 0,
from" which
cp = — ax.
The function
f(z) = — a z + Ae~ik* = - a (x + iy) -f Ae-^+W
gives
(p + ^ = — a (% + iy) 4- Aeky (cos kx — i sin kx),
134) (p = — ax + Aekv cos kx,
if/ = — ay — Aeky sin kx.
When y = — cx> this makes (p = — ax, as required. The free surface
of the water being composed of stream lines is represented by one
of the lines ^ = const, and if we take the origin in the surface its
equation is consequently
135) ay -f- Ae^sinkx = 0,
which shows that y is a periodic function of x with the wave-length
I = -^- The longer the wave-length, that is the smaller k, the more
1) Rayleigh, On Waves. Phil. Mag. I, pp. 257— 279, 1876. Scientific Papers,
Vol. I, p. 261.
WEBSTER, Dynamics. 34
530 XI. HYDRODYNAMICS.
nearly does the exponential reduce to unity and the more nearly is
the profile a curve of sines. The velocity is given hy
136) £2 = u2 + v*,
where
u = fl = v = ~ a-Ake^smJsx,
137) a *
I? = = - > = Ake**coakx,
138) #2 = a2 + ^fcV** + 2^afce** sin lex.
So far all our work has been kinematical. The relation to
dynamics is given by introducing the equation 33) for steady motion,
and at the surface putting jp = 0, and making use of the equation 135),
140) gy + i {a2 + A*k*#*y - 2a*ky} = C.
Since the surface passes through the origin, putting y = 0 we obtain
C=1-{a* + A*V},
inserting which gives
141) (g - a2fy y + ± AW (e^ - 1) = 0.
This equation can be only approximately fulfilled, but if the height
of the waves is small compared with the wave-length, so that 2 ky
is small, developing the exponential and neglecting terms of higher
order than the first in ky we have
giving the equation connecting the velocity and wave-length
142) g - a2k + AW = 0.
If ky is small the equation of the surface 135) is approximately
143) y = -- sin kx
so that the maximum height of the waves above the origin is T$ = —-
Inserting the values of the height and wave-length in equation 142)
it becomes
8f2«/. 4:*2.BM
144) a {TV --^1 ~ *
an equation connecting the wave-length, height and velocity. For
198] WAVES IN DEEP WATER. 531
waves long enough, in comparison with their height to neglect — ,2 ->
we have
145) . «»=.£
If s is the height from which a body must fall to acquire a velocity
equal to the wave - velocity, since a2 = 2gs, the equation becomes
146) I = ±xs,
accordingly the velocity of propagation of long waves in deep water
is equal to the velocity acquired by a body falling freely from a
height equal to one -half the radius of a circle whose circumference
is the wave-length.
In order to study the motions of individual particles of water
let us now impress upon the motion given by 137) a uniform velocity a
in the X- direction. Equations 137) now give the motion with respect
to moving axes travelling with the waves, so that in order to obtain
the motion with respect to fixed axes we have to add a to the u
of 137) and replace x by x — at, obtaining
u = — A~ke*y sin k (x — at],
147)
ij = Akeky cosk(x — at),
for the equations of the unsteady motion of the actual wave -propa-
gation. For the velocity of a particle we have
148) q = -\/u2 + v2 = AJce*y
showing that the velocity decreases rapidly as we go below the
surface, so that for every increase of depth of one wave-length it is
reduced in the ratio e~27t = .001867. If the displacement of a particle
which when at rest was at x, y is |, 17 we have
-FT = — A~kekv sin~k(x — at),
149)
if we neglect the small change of velocity from x,y to x -\-%,y -\- r},
so that we obtain by integration
| =
1 F\A\
il =
Thus each particle performs a uniform revolution in a circle of
2 it, >L
radius Beky in the periodic time y— = — • We thus see how the
ka a
motion is confined to the surface layers. The direction of the motion
in the orbit is such that particles at the crest of the wave move in
the direction of the wave -propagation, those at a trough in the
opposite direction.
34*
532
XI HYDRODYNAMICS.
Let us now discuss the form of the wave -profile 135) when the
restriction that the height of the waves is small in comparison with
the wave-length is removed. The equation of the surface is
This may be conveniently done by means of a graphical construction,
Fig. 168. Let us
construct two
curves, with the
coordi-
the
runnng
nates X, Y,
first the logarith-
mic curve
Fig. 168.
X = e*Y
which must be
and the second the straight line X = — p . 7
B smkx
separately constructed for each value of x. At the intersection of the
line and curve , we have
Y+Be*Y sin 7^ = 0,
so that the value of Y thus obtained may be taken for the y coordinate
of the wave -profile with the abscissa x. As x varies, the line swings
back and forth about the X-axis, and we see that when sin is
positive there is one intersection of the line and curve, while if sin ~kx
is negative there are two, giving two values of y, both positive.
Any positive y is greater in absolute value than the corresponding
< / negative for the
\ / / symmetrical posi-
\ I \ / tion of the line.
/ / Thus the unsym-
\ / metrical nature of
"'V_..X trough and crest
is made evident.
Beginning with
_^> -\^^ ^ ^\^^ x = 0, the two
'f^^ ^^"-~~^_ _—- — ^^ ^~ values of y are
Fig- 169. one zero, the other
infinity, and as x increases, y has a single negative value. When x=-^- = —>
y is again zero and infinity, and as x increases the two values of y, both
Q
positive approach each other until y = — A, then recede until y = L
The form of the curve as constructed in this manner is shown in
Fig. 169, the lower branch representing the wave -profile. If B is
greater than a certain quantity the values of y between certain limits
198]
HIGHEST WAVES IN WATER.
533
are imaginary. This limiting value of B is that which makes the
highest position of the straight line, for which sin kx = — 1, tangent
to the exponential curve. We then have
j -y V -y -|
jy = kX for the curve, equal to yr= ^- for the line,
from which
The upper and lower branches of the curve 151) then come together,
and the wave -profile has an angle. Waves cannot be higher than
this without breaking. By differentiation of 151) we find for the
summit, -^- = + 1, so that the angle between the two sides of the
wave is a right angle (Fig. 170). As a matter of fact, before
the waves are as
high as this, the \
equation 141) is \ /
no longer satisfied \ /
with sufficient \ /
approximation
for the waves to
have the form in
question. By an
elaborate system
of approximation,
Michell1) has
shown that the highest waves have a height .142 A, while the equa-
tion 151) gives .2031. It was shown by Stokes2) that at the crest
the angle was not 90°, but 120°, as follows.
In the stationary wave, in order to have an edge, the velocities u
and v for a particle at the surface must both vanish together, for
if v alone vanishes, there will be a horizontal tangent. Consequently,
if we place the origin at the crest, equation 139) becomes
Fig. 170.
gy
o.
But if we represent the surface by a development of the form of
equation 128), on account of symmetry there will be only sine terms,
and if in the neighborhood of the origin we retain only the most
important term, we may put
1) Michell, The highest Waves in Water. Phil. Mag. 36, p. 430, 1893.
2) Stokes, On the Theory of Oscillatory Waves. Trans. Cambridge Philo-
sophical Society, Vol. VIII, p. 441, 1847. Math, and Phys. Papers, Vol. I, p. 227.
534 XL HYDRODYNAMICS.
152) gp = Arn sin no, ty = Arn cos ncc>,
o being the angle measured from the vertical. We have for the
radial velocity
G OP .
qr = -£- = Anrn~-1smn(o,
and if a is the inclination of the surface to the vertical at the crest
q = Anrn~1smn a. But we have g2 = — 2gy = 2gr cos a and accord-
o
ingly 2 (n — 1) = 1, n = —- Also as in 129), cos na = 0. Thus
The problem of waves in water of finite depth may be treated
in a similar manner, by putting instead of 134),
cp -f ty = — az + Ae~ikz -f jBe*'**,
qp = — «# -f (^le*y -f Se~ky) coshx,
153) il> = — ay — (Aekv - Be~*y) sin A; a;,
w = — a -
v=
If the depth is h, we must have v = 0 for y = — h9 giving
Ae~kh = Bekh.
Calling this value C, we have
154) ^ = -ay- <7(e *(*+*) - e-*(*+y)) sinA;^ = 0,
as the equation for the wave -profile. For the first approximation,
for waves whose height is small compared to their length, replacing
eky,e~ky by unity, we have
155) ay = — C(ekh — e-kh} • sin,
and neglecting (GY&)2,
156) u2 + v2 = a2 + 2CW (ekh -f e~k/<) sin kx.
Thus the surface equation 139) becomes
157) const = ^--^ (ekh - e~kh) sin Jcx
^
a
-f a2
which is satisfied by
158) a~k(e]
giving the velocity
159)
198, 199] WAVES IN SHALLOW WATER. 535
If h is infinite this reduces to 145), while if the depth is very small
with respect to the wave-length, it reduces to a? = gh. Accordingly
long waves in shallow water travel with a velocity independent of
their length, being the velocity acquired by a body falling through
a distance equal to one -half the depth of the water. Consequently
the resultant of such waves of different wave-lengths is propagated
without change, contrary to what is the case in deep water.
Changing to fixed axes, we have for the running wave
- *(A0 - -*(A in & (x _ af),
and by comparison with 147), 150), we find that the particles
describe ellipses with semi -axes equal to
161) G (e*(A+y)-|- e-*(A+?)), C (e*(A+y)-_ g- *(*+?)).
If we consider the resultant of two equal wave -trains running
in opposite directions, we have
sfc(# — at) -f cos ~k (x -f at)]
= 2 (7 («*<*+*>+ e-^+y^cosJcxcosJcat,
sin k (x - at} -I- sin k(x + at)]
The equation of the profile is now of the form, y is equal to a
function of x multiplied by a function of tf, so that the profile is
always of the same shape, with a varying vertical scale. Such waves
are called standing waves, and we see them in a chop sea. The
difference between them and the stationary wave in a running stream,
with which we began, is very marked, as here every point on the
surface oscillates up and down, while there the water -profile was
invariable both as to time and place.
199. Equilibrium Theory of the Tides. We shall now
briefly consider some aspects of the phenomena of the tides, the
general theory of which is far too complicated to be dealt with here.
The earliest theory historically is that proposed by Newton, which
supposes that the water covering the earth assumes, under the attraction
of a disturbing body, the form that it would have if at rest under
the action of the forces in question. This so-called equilibrium
theory, which neglects the inertia of the water, belongs logically to
the subject of hydrostatics, but will be now treated. If U denote
the potential of gravity, including the centrifugal force, as in § 149,
we have, as there, for the undisturbed surface of the ocean,
536 XI. HYDRODYNAMICS.
163) U (r0, if>, ,
,g>)-U (r0, ^ 9) + F = const, = C.
But if we put h = r — r0, h is the height of the tide, and being
small with respect to the radius, we may put
166) U(r,^ri-U(rQ,^,ri = h™,
giving
167) v-c=-hj?-
But g = — ?-*-> as in § 149, so that we obtain for the height of
the tide
168) h = r°.
We may determine the constant in 168) by the consideration
that the total volume of the water is constant. If dS is the area
of an element of the earth's surface, the total volume of the tide
above the surface of equilibrium must vanish, giving
169) 0=hdS, VdS = cdS, V=C,
where F is the mean value of the disturbing potential over the earth's
surface. Now we have found in § 150, equation 154), the value of
the potential of the tide - generating forces,
170) F-£g(3cos»Z-l),
where Z is the zenith-distance of the heavenly body at the point in
question. If we refer other points on the earth's surface to polar
coordinates with respect to this point and any plane through it,
with coordinates Z, <&, we have
2rt TI
C CvdS = ™ Cd® f@ cos2 Z - 1) sin Z dZ = 0,
0 0
so that the mean of F vanishes. Accordingly we have
171) fc-.(3oo.»Z-l).
199] EQUILIBRIUM THEORY OF TIDES. 537
This equation shows that the tidal surface is a prolate ellipsoid of
revolution, with its axis pointing at the disturbing body.
Let us now express cos Z in terms of the latitude iff of the point
of observation and of the declination d and hour -angle H of the
disturbing body, which for brevity we shall call the moon. If we
take axes in the earth as usual, with the XZ- plane passing through
the point of observation and measure H from this plane, we have
for its coordinates and those of the moon respectively
r cos iff, D cos d cos H,
0 ; D cos d sin H,
r sin ijt, D sin d ,
from which we obtain the cosine of the angle included by their radii
cos Z = cos iff cos d cos H -f sin if> sin d.
Squaring this, replacing cos2 H by -_ - (1 4- cos 2-ET), cos2 iff cos2 d by
(1 — sin2 i{i)([ — sin2 d), we easily obtain
3 cos2 Z — 1 = -~ [cos2 d cos2 ty cos 2H + sin 2 d sin 2^cosH
(l-3sin2(?)(l-3sin2ij/n
8^ — J-
Inserting this in 171), replacing g by its approximate value -Tr and,
as we have already done, neglecting the attraction of the disturbed
water, we have the equation for the tide,
172) * = ^^ [cos2 d cos2 ^ cos 2H + sin 2 d sin 2 # cos H
(1-3 sin2 d) (1 - 3 sin2i|;)-|
~~3~~ J
The first term in the brackets, containing the factor cos 2 JET,
where H is the moon's hour -angle at the point of the earth in
question, is periodic in one -half a lunar day, consequently this term
has a maximum when the moon is on the meridian, both above and
below, low water when the moon is rising or setting. The effect of
this term is the semi-diurnal tide, which is the most familiar, with
two high and two low waters each day. This tide is a maximum
for points on the equator, where cos2 ty = 1, and for those times of
the month when cos2 d = 1, that is when the moon is crossing the
equator. These are the so-called equinoctial tides.
The second term, containing the factor cosJS, is periodic in a
lunar day, and gives the diurnal tide. This gives high water under
the moon, and low water on the opposite side of the earth. On the
side toward the moon, these two tides are therefore added, while on
538 XL HYDRODYNAMICS.
the opposite side we have their difference. Consequently, at any
point, the difference of two consecutive high waters is twice the
diurnal tide. This difference is generally small, showing that the
latter tide is small. It vanishes for points on the equator, and at
the times of the equinoctial tides.
The third term, which vanishes for latitude 35° 16', does not
depend on the moon's hour -angle, but only on its declination. This
declinational tide, depending on the square of sind, has a period of
one -half a lunar month.
Beside the tides due to the moon, we must add those due to
the sun, for which the factor outside the brackets in 172) is some-
what less than one -half that due to the moon. The highest tides
therefore occur at those times .in the month when the sun and the
moon are on the meridian together, namely at new and full moon.
These are known as spring -tides. The lowest occur when the moon
is in quadrature with the sun, and the lunar and solar tides are in
opposition. These are known as neap-tides, and occording to this
theory would be only one -third the height of the spring -tides. The
greatest spring- tides would be those in which the moon was on the
equator, or the equinoctial spring- tides. Now it is found that, instead
of this, the high tides come about a day and a half later. Consequently,
although the equilibrium theory indicates to us the general nature
of the different tides to be expected, it does not give us an accurate
expression for their values. Roughly speaking we may say that the
tides act as if they were produced as described by the action of the
sun and moon, but that the time of arrival of the effects produced
was delayed.
A correction was introduced into the equilibrium theory by
Lord Kelvin, to take account of the effect of the continents. For if
the height of the tide were given by the equation 171), removing
the various volumes of water in the space actually occupied by land
would subtract an amount of water now positive, now negative, so
that the condition of constant volume would not be fulfilled. In
order that it still may do so, the integral 169) is to be taken only
over those parts of the earth's surface covered by the sea, The
value of V is then not zero. The effect of this is to introduce at
each point on the earth's surface change of time of the arrival of
each tide, varying from point to point. The practical effect of this
correction is not large.
20O. Tidal Waves in Canals. In the dynamical theory of
the tides, taking account of the inertia of the water, we have the
problem of the forced oscillations of the sea under periodic forces.
As a simple example illustrating this method we shall consider waves
199, 200J CANAL THEORY OF TIDES. 539
in straight canals. Let the motion be in the plane of XY, as in
§ 198, and let Ji, the depth of the canal, be small in comparison
with the wave-length. We shall suppose the displacements of all
the particles, with their velocities and their space -derivatives, to be
small quantities whose squares and products may be neglected. We
shall also neglect the vertical acceleration, so that the equation for y
is that of hydrostatics, giving the pressure proportional to the distance
below the surface. If the ordinate of the free surface is h -\- y, this
gives
173) P
174) = '
ox y * ox
while the equation of motion, the first of equations 6), is
175) |» x_i|£.
Ot Q OX
Combining these two equations, we have
17«\ cu v dr]
176) w=x~^'
Q
and if X is independent of yy since ^ is also, this shows that u
depends only on x and t, or vertical planes perpendicular to the
XY plane remain such during the motion.
Integrating the equation of continuity
du cv A
Wx + Wy="(}>
with respect to y from the bottom to the surface,
1 nn\ tdu ^ /7 . N du
1<7) v- -J^dy- -(h + r,)^,
or approximately, at the surface,
-, rro\ $n 7 ^U
178) v = Tt=-]lTTx
O £
Now putting u = 3! ? the equation of continuity 178) becomes,
ot
1 7Q^ ^^
= ~
and on integration with respect to the time,
180) 1~-*H-
Substituting in 176) we have for the horizontal displacement
540 XI. HYDRODYNAMICS.
If there is no disturbing force, X = 0, and we have the equation
for the propagation of free waves, which we might have used in
order to obtain the results of § 198, for instance it is satisfied by
equations 150) if we put a2 = gh. This is the same equation as we
had in § 46, equation 109), for the motion of a stretched string,
and the standing waves of 162), § 198, putting y = 0, are the normal
vibrations of equation 115), § 46. The general solution of equation 182)
is obtained in the next section, for the present it is sufficient to
consider the wave already obtained which advances unchanged in
form with the velocity a. We have then, in the case of an endless
canal encircling the earth, the curvature of which we may neglect,
the case of a free wave, running around and around, without change,
so that at any point, the motion is periodic in the time — ; where I
is the length of the endless canal. We thus have a system with
free periods, and when we consider the action upon it of periodic
disturbing forces, we may expect the phenomena of resonance, as
described in Chapter Y.
Let us now suppose the canal coincides with a parallel of
latitude, and that x is measured to the westward from a certain
meridian. We then have for the horizontal component of the
disturbing force
where V is given by 170), and H, the hour -angle of the moon at
the point x, is
x dV cV 1
183) H=at— —> so that — ~— = ^^ - —>
rcosi|) ex dHrcoaip
03 being the angular velocity of the moon with respect to the earth.
We accordingly find X to be composed of two terms each of the form
where for the semi-diurnal part
184) A = — -jjj- cos2 ^ cos tyj m — 2&, fr = —
Introducing this into the equation 181),
185) r4 = &2 o-| — ^.sin(w£ — Jcx).
Ot VX
we may find a solution
200] .DIRECT AND INVERTED TIDES. 541
186) % = Bs
where by insertion in 185) we find
From 180) we obtain
188) 77 = ^irr£i cos (m t - Jcx).
The coefficient of the cosine is positive or negative according as ok
is greater or less than m, so that we have, according to circumstances,
high or low water under the moon. In the former case, the tides
are said to be direct, as in the equilibrium theory, in the latter they
are inverted. But — '- is the ratio of the time period of the force,
m
or half a lunar day, to the time required for a free wave to travel
half around the earth, and the tide is direct or inverted according as
this is greater or less than unity. Equation 188) is the analogue of
equation 50), § 44. Inserting the values of the constants in 188)
we find that the canal theory gives the height of the tide as given
by the equilibrium theory in 172) (which we also obtain by putting
m = 0), multiplied by the factor
-f-Y
(ak)
exactly as described for the system with one degree of freedom on
page 155. If we introduced into our equations a term giving the
effect of friction we should obtain a change of phase, as in § 44, of
amount other than a half -period, or inversion.
In order to determine the directness or inversion of the tides,
let us insert the values of m, k from 184) in 188), by which we
find that the tides are direct or inverted according as we have the
upper or lower sign in the inequality
189) 0&^r8a>2cos2V>.
Supposing the lunar day to be 24 hours, 50 minutes, the earth's
circumference forty million meters, we find at the equator the critical
depth, determining the inversion, to be 20.46 kilometers, or 12.7 miles.
As the depth is less than this, the tides are inverted. For any depth
less than the critical depth, there will be a latitude beyond which
the tides will be direct. Accordingly we see that even if we consider
the ocean to be composed of parallel canals separated by partitions,
the tides will be very different in different latitudes, so that if the
partitions be removed, water will flow north and south. We thus
obtain an idea of the complication of the actual motion of the tides.
542 XL HYDRODYNAMICS.
By introducing the complete expressions for the accelerations with
respect to revolving axes, given in § 104, and applying the principles
of forced oscillations, we obtain the more complete theory given by
Laplace.
2O1. Sound -Waves. Let us now consider the motion of a
compressible fluid which takes place in the propagation of sound.
In the production of all ordinary sounds, except those violent ones
produced by explosions, the motion of each particle of air is extremely
minute. We shall therefore suppose that the velocity components
u, v, w and their space derivatives are so small that their squares
and products may be neglected. Let us put
190) p = p0(l-M),
where QO is a constant and s is a small quantity, of the same order
as the velocities, called the compression. From the equation of
•continuity we have
1Q1x _ du cv dw _ 1 d$ _ 1 ds
-fa^d^^fa- ~~Q~di~ ~1+10*'
or neglecting the product of s and its derivative,
-t f\c\\ ds
192) "=-**•
In order to calculate P, we have, since the changes in Q are small
193) dp = a2dQ = a2QQds,
where a2 is a constant representing the value of the derivative J-
for Q = QO, the density of the air at atmospheric pressure. We
therefore have
194) P _ =-«• log (1 +.) = ««,,
to the same degree of approximation.
Neglecting small quantities the equations of hydrodynamics 6)
become, when there are no applied forces,
o o
CU 9 08
-or = — a 3— 7
dt dx
dv 9 ds
195) - = — a?2->
dt dy
dw o ds
—- = — a2-^-)
dt cz
with
192) «=-f,
Differentiating the equations respectively by x, y, z, adding and
observing the definition of 6, we obtain
200, 201, 202] SOUND-WAVES. 543
196) =
and differentiating 192) by t and combining with this
197) *J
Since the motion is assumed to be irrotational, introducing the velocity
potential into equations 195) they become the derivatives by x, y, z
respectively of the equation
198) Si—''-
Differentiating by t, making use of equation 192),
199) j£ = - - a2 d~ = a2 6 = a*4
plf the motion is sometimes called a pulse. A pulse is
none the less a wave.
Thus the general solution of equation 201) represents two plane
waves propagated in opposite directions with the same velocity a.
The velocity of sound a = y -JJT depends upon the elasticity of
the air and was calculated by Newton, assuming that the process
was isothermal, using Boyle's law. As this was found to give results
not agreeing with experiment Laplace suggested that the compression
was adiabatic, the vibrations being so rapid that the heat generated
did not have time to .flow from the heated to the cooled parts. Thus
the constant K, equation 17) § 178 representing the ratio of the two
specific heats of the air is introduced. The velocity of sound gives
one of the most accurate ways of determining this ratio sc.
The velocity of the particle of air is obtained by
C)f\Q\ d = i{F1(r-aO + F,(r + at)\,
of which the first term represents a wave proceeding outwards, the
second one proceeding inwards, the magnitude however varying
according to the factor — •
For a periodic solution representing a simple tone proceeding
from a single point -source we may take
218) (p = — -- — cosJc(at — r).
*
The physical meaning of the constant A is obtained as follows. Let
us find the volume of air flowing in unit of time through the surface
of a sphere with center at the source. We will call this the total
current,
219) I
= A {cos ~k(at — r) — kr sin ~k (at — r)}.
Accordingly when r = 0 we have I = Aeoshat and A, the maximum
rate of emission of air per unit of time, is called the strength of the
source, agreeing with the definition of § 196.
In order to obtain the activity of the source, that is the rate
of emission of energy per unit of time, we may find the rate of
working of the pressure at the surface of a sphere, as explained in
§ 188,
220) P =
In order to find p, we have, if pQ is the undisturbed atmospheric
pressure, by integration of 193), and by 198),
35*
548 XL HYDRODYNAMICS.
&£\. j p — _2^Q == ^ ^o^ == — ?o ~
from which we obtain
222) P =
This contains a part which is alternately positive and negative, and
also one which is always positive. If we seek the mean value of P
throughout the period, that is
T
P = — f Pdt, T=~,
o
we easily find, since the mean of cos#, sin#, cos # sin #, is zero,
while the mean of sin2^ is —>
223) P = A*f**a
which is independent of the radius, as it should be. The mean
energy -flow per unit of time and per unit of area of the sphere is
which is a measure of the intensity of the sound. Tihs decreases
as the inverse square of the distance. In order to give an idea of
the extremely small dynamical magnitudes involved in musical sounds,
it may be stated that measurements made by the author1) showed
that the energy emitted by a cornet, playing with an average loudness,
was 770 ergs per second, or about one ten -millionth of a horse-
power, while a steam -whistle that could under favorable circumstances
be heard twenty miles away emitted but - — ? or one -sixtieth
of a horse -power (see note, p. 153).
2O5. Waves in a Solid. The equations of motion for an
elastic solid are obtained from the equations of equilibrium 144), § 175
by the application of d'Alembert's principle in the same manner as
the equations of hydrodynamics were deduced from those of hydro-
statics. It will be convenient here to revert to the notation of
Chapter IX where u, v, w and 6 refer to displacements rather than to
velocities. Applying d'Alembert's principle we thus obtain
225) p (X - g) + (I + ,*) fl + ^u = 0, etc.
1) Webster, On the Mechanical Efficiency of the Production of Sound.
Boltzmann- Festschrift, p. 866, 1904.
204, 205, 206] WAVES IN SOLID. 549
If there are no bodily forces we have the equations of motion
da
226) e=(* +
d*w ,. . Ndff .
V ~W = ( + **' Tz + P w'
Differentiating respectively by x, y, z and adding we obtain
227) e = (i
which is the equation for the propagation of wave -motion, the
dilatation being propagated with a velocity b = y- — -• Taking
the curl of equations 226) we have
228)
Thus the components of the curl are propagated independently, each
with a velocity a—y— - The velocity of the compressional wave
which is unaccompanied by rotation depends upon the bulk modulus
and the modulus of shear. The velocity of the torsional wave which
is unaccompanied by change of density depends only upon the mo-
dulus of shear. The general motion of an elastic body is a com-
bination of waves of compression and of torsion. The wave of
torsion is that upon which the dynamical theory of light is founded.
Inasmuch as p vanishes for a perfect fluid no wave of torsion is
propagated, so that the luminiferous ether must have the properties
of a solid and not those of a fluid.
2O6. Viscous Fluids. We have now to consider a class of
bodies intermediate in their properties between solids and perfect
fluids, namely the viscous fluids. By definition a perfect fluid is one
in which no tangential stresses exist. We have then
229) Xx = Y, = Z, = -p, X, = Y2 = ZX = 0.
In a fluid which is not perfect no tangential stresses can exist in a
state of rest, but during motion such stresses can exist. While in a
solid the stresses depend on the change of size and shape of the
small portions of the solid, in the case of a viscous fluid the stresses
550 XL HYDRODYNAMICS.
depend on the time -rates of change, that is on the velocities of the
shears, stretches, and dilatations. The simplest assumption that we
can make is that the stress -components are linear functions of the
strain -velocities. The fluid being isotropic, considerations regarding
invariance bring us to precisely similar conclusions to those we
reached in § 175, so that to the stresses of equations 229) for a
perfect fluid are added stresses given by equations 142), § 175, A and p
being constants for the fluid, and u, v, tv, 6 now denoting velocities,
instead of displacements, returning to the notation of this chapter.
(We put P = 0, since these additional terms vanish with the velocities.)
We thus obtain
Z2 = -p + ^ + 2ti~z
230)
which are of the same form, with a different meaning, as 142), § 175.
If the fluid is incompressible we find, putting 6 = 0,
Xx + Yy + Z,= - 3p,
and assuming that this holds also for compressible fluids we must have
231) 3^ + 2^ = 0.
a
Replacing A by its value - — jt, we find for the forces, as in § 175, 144),
v dp . 1
232)
r, 3p . 1
eZ-U+ 3
which are to be introduced into the equations of hydrodynamics 6).
Thus we obtain the general equations, putting — = v,
du , du , du , cu v ds ^ 1 dp
^rr-h^K -- h^o -- h^o -- -^ o -- V^U = X -- K^-J
dt dx ' dy ' dz 3 0x Q dx
rtoo\ ^^ dv dv t dv v da T;r 1 dp
233) - H-MO — h^^-f^o- - vdv = Y -- 7p->
' ot dx dy dz 3 dy Q dy
dw . dw . dw . dw v ds ^,1 dp
-r-\-U~ -- h^o -- h^o -- ^ - —vAw = Z -- r~?
dt ' dx ' dy ] dz 3 dz Q dz
206] VISCOUS FLUIDS. 551
which reduce to 6) when ^ = 0. The coefficient p is called the
viscosity of the fluid, and its quotient by the density, v, is called by
Maxwell the kinematical coefficient of viscosity.
The equations 233) are too complicated to be used in all their
generality. We shall here consider only the case of incompressible
fluids, for which the terms in 6 vanish. If we form the equation
of activity as in § 188, we obtain beside the terms in the first integral
of 29) the additional terms
~ P I I I
4-
which by Green's theorem may be converted into
dv dw
u\2, /2»\2 , /3sA3, /M2 ,
w + u + y + u +
If the integration be extended to a region where the liquid is at
rest, say the surface of a containing solid, where the liquid does not
slip, the surface integrals vanish, and the volume integrals give a
positive addition. That is to say, the applied forces have to do an
amount of work over and above that going into kinetic and potential
energy, and this work is dissipated into heat. If there are no applied
forces, the energy of the fluid is dissipated, and it will eventually
come to rest.
In order to find simple solutions of our equations, we may deal
either with steady motion, or with motions so slow that we may
neglect the terms of the second order in u, v, w and their derivatives.
Let us first consider steady motion. The simplest case is uniplanar
flow parallel to a single direction, or as we may call it, laminar flow.
If we take
noA\ du dp
234) 0 = v = w= -75- = -^-9
} dz dz
the equation of continuity gives
235) *£ = 0.
If there are no applied forces, equations 233) reduce to
OQ£\ d*u dp dp A
236) ^Q-2=:r^ ^ = ^
^ cy* ox oy
Since u depends only on y and p only on x, this equation cannot
hold unless each side is constant. Accordingly
dzu dp 1 a 9
^72==a==^' M = c + &«/-f-¥ -y2, p = d + ax,
552 XI. HYDRODYNAMICS.
where a, Z>, c, d are constants. If we determine them so that the
velocity vanishes for planes at a distance +h from the X-axis, we have
238) u = ^(f-V).
The amount of liquid that flows through such a laminar tube per
unit of width parallel to the Z-axis is accordingly
239)
— h
and for a length of tube I the difference of pressures at the ends is
240) • A-ft-aJ, 3=££,
so that the flow is proportional to the difference of pressures at the
ends and inversely to the viscosity.
For the practical determination of viscosity, we may take the
almost equally simple case of cylindrical flow, where the velocity has
everywhere the same direction, and depends upon the distance r from
the axis of a circular tube, at the surface of which it is at rest.
If we put u = v = 0 we have the equations of motion and of
continuity
o^i\ dp
241)
and since w depends only on r the first becomes
C*AO\ , dp
242)
where a is a constant as before. This equation is integrated as in
§ 182, 58'),
243) w = ^-r2 + fclogr -f c.
Since w is finite when r = 0 we must have b = 0 and if w vanishes
for r = R we obtain
244) „„£(,.
For the flow we find
ctAc.\ s\
245) Q =
0
This method was invented by Poiseuille1) for the measurement of
1) Poiseuille, Eecherches experimentales sur le mouvement des liquides dans
les tubes de tres petits diametres. Comptes Rendus, 1840 — 41; Mem. des Savants
Strangers, t. 9, 1846.
206] DETERMINATION OF VISCOSITY. 553
the viscosity of fluids. His verification of the proportionality of the
flow to the fourth power of the radius of the tube has been taken
as a proof that the liquid does not slide when in contact with a solid.
As another example of steady flow let us consider uniplanar
cylindrical flow, in which each particle moves in a circle with velocity
depending only on the distance from the axis, as in the case of the
lubricant between a journal and its bearing. Each cylindrical stratum
then revolves like a rigid body, which requires
246) u = — &y, v = ox,
where co depends only on r — "J/#2+ y2. We then find
du xy do 3u y* do
— = -- -=—y ~ — = — O --- -3—?
cx r dr cy r dr
' dv . X* dco dv xy dm
x
.
= <*> H ---
-5—' o~ = — ~^-
r dr oy r dr
and, most easily by the application of equation 86), § 141, and by
the expression of z/co in terms of r,
d(o\ ~ y
~
- -=- — 2- -=-t
r drj r dr
dco\ . rt x dco
o) . 1 dco\ . rt x
r + - -J-) + 2-
* ' r dr) ' r dr
Thus the first two of equations 233) become
/d2a . 3 d&A 1 x dp
-f vyl-j-j- H — -3-1 — -- - i
» \dr* r dr) Q r dr
249)
2 ( I ^ \ * 2/ **.P
\c?r2 r dr/ Q r dr
Multiplying the first by y} the second by x and subtracting,
250) 4^ + - ?- = 0,
a?*2 r ar
a differential equation whose solution is
251) « = £ + &.
Determining the constants so that CD = 0 for r = R± and o> = & for
r = R*,
252) „,= f-V Jj^-'.-i|.
1 2 \ J
Multiplying equations 249) by x and y respectively, and adding, we
have to determine PJ
253) £eo2r = ^-
For the stresses we obtain, using equations 230),
v I 0 xydo)\ (x*-y*
Xn = ( — p — 2p -^ ^-J cos (nx) H- it I - •* -rr I cos
554 XI. HYDRODYNAMICS.
n*A\ T^ /x*—y*da\ / \ , / «r» xti d(o\ , N
254) Yn = ii (—^L ^j cos (nx) + (- # + 2jt -^ ^J cos (wy),
Zn = — p cos (w#) = 0.
This shows that there is a normal pressure p, together with a tangential
stress which we obtain by resolving along the tangent,
255) T = Yn cos (nx) - Xn cos (ny)
dofo;2— w2, 9 9 \ , ±xy / \ / >1
~ ^dr \ — r~ (COS "~ COS ^' "^ -- COS v1*) cos (w^) I
and since cos (nx) = —^ cos (ny) = —^
The moment of the tangential stress on the cylinder of radius r and
unit length, is accordingly
257)
We may accordingly use this method to determine the viscosity, as
is in fact done in apparatus for the testing of lubricants. We see
that if the linear dimensions are multiplied in a certain ratio, the
moment is increased in the square of that ratio. We also see that
the moment of the force required to twist the cylinder is independent
of the pressure p, which contains an arbitrary constant, not given
by the equation 253), but depending on the hydrostatic pressure
applied at the ends.
Let us now consider some simple cases where the flow is not
steady, limiting ourselves to the case of small velocities, so that the
terms in 233) involving the first space derivatives, being of the second
order, are negligible. Let us once more consider laminar flow, defined
by equations 234), 235). Let us also put p = const. Instead of 236)
we now have for the first of equations 233),
258) ^==vd^.
dt fly*
This equation is the same as that which represents the conduction
of heat in one direction. Let us first consider a solution periodic in
the time, such as may be realized physically by the harmonic small
oscillation in its own plane of a material lamina constituting the
plane y = 0, along which the liquid does not slip. We may take as
a particular solution
which inserted in 258) gives
n = vm
206] SLOW MOTIONS. 555
If this is to be periodic in t, n must be pure imaginary, say
n = ip.
Then we have
and
of which complex quantity both the real and the imaginary parts must
separately satisfy the equation 258), when multiplied by arbitrary
constants. Let us accordingly take
259) u = | A cos (pt — y -j- • y) + J5sin (pt —
\ \ i A r / \
This represents a wave of frequency ~^ and wave-length 2jrl/-
travelling with velocity y%vp, which as we see varies as the square
root of the frequency. Unlike our waves in perfect fluids however
it falls off in amplitude, being rapidly damped as we go into the
fluid, being reduced in the ratio e~ ** = — in each wave-length.
Thus such motions are propagated but a short distance into a fluid.
In a similar manner the absorption of light by non- transparent media
is explained, the ether there having the properties of a viscous solid.
If we treat the equations 233) in the same way as we did 27)
in obtaining equation 57), § 191, we obtain instead the following,
d /|\ &du ndu £ du <.
di (j) = i fa + 7 ^ + -Q dJ + v * «>
260)
dt \Q/ Q dx g dy
d £\ g dw r 3w
Under the circumstances of slow motions these also reduce to
261)
Thus we see that the three components of the vorticity are propagated
independently, each according to the equation for the conduction of
556 XL HYDRODYNAMICS.
heat. The example just treated is an example of this, for we find
at once
2 f) QJ
and the vorticity is propagated like the velocity.
As a final example, let us consider a case of laminar motion in
which u, as a function of y, has a discontinuity, this having an
important application to the theory of thin plane jets and flames,
including sensitive flames.1) We will suppose that at the time t = 0
for y < 0 u has a certain constant value, and that for y > 0 it has
a different constant value. It is easy to see that this is equivalent
to supposing that there is no vorticity except in an infinitely thin
lamina at y = 0. For we have
s s
/-* /-*
263) /£^ = --l / ^dy = ~(u±-u,)
/ i ••> i/ 2 / c i/ 2 ^ * •'
t/ t/ y
— e — c
where u± is the velocity on one side, u2 that on the other of the
layer of thickness 2s. Now if the thickness decrease without limit,
while g increases without limit, the integral may still be finite, as
we shall suppose.
We have then to find two solutions u and f of equation 258),
so related that g = — — -^- • Let us put s = — '*— > and try to find a
particular solution that is a function of s alone. We have
du du ds _ 1 du y
dt ~~ ds dt 2 ds }/i*
du du ds du I
d^==~ds^~'ds'y^)
d*u _ 1 d*u ds _ £ d*u
dy9 yt ds* dy t ds*
so that our equation becomes the ordinary differential equation,
1) Rayleigh, On the Stability, or Instability, of certain Fluid Motions
Proc. London Math. Soc., xi., pp. 57— 70, 1880. Scientific Papers, Vol.1, p. 474.
206] SHEET OF DISCONTINUITY. 557
The integral of this equation is given by
267) los<^ = -£- + const., £ = ««'£, »
The last indicated quadrature cannot be effected except by development
of the integrand in series, but if we take for the lower limit the
value zero, we may express u in terms of the so-called error -function,
occurring in the theory of probability,
268)
Tables of the values of- Erf(x) have been calculated, and are found
in treatises on probability. (Lord Kelvin reprints one such on p. 434
of Vol. 3 of his collected papers.) Since the integrand is an even
function of x, it is evident that Erf(x) is an odd function of its
upper limit x. It may be easily shown that the definite integral
between zero and infinity has the value -^p so that putting x* = -j—>
and adding a constant, we have
269)
This determination of the constants makes, for all positive values
of y and for t = 0, u = u^ (the upper
limit being -f oo), and for all negative
values u = u2, thus giving the dis-
continuity required at y = 0. For all
other values of t however, no matter
how small, the values from the negative
side run smoothly into those on the
positive, showing how the discontinuity
is instantly lost. This is shown in Fig. 171, in which successive
curves show values of u at times equal to 1, 2, 3, 4, 5, 6 times —
Differentiating by the limit, we find
270) fs»_4.|«.
558 XL HYDRODYNAMICS.
which is infinite when t = 0, - = = 0, as we supposed, but which
immediately drops to a finite value, and, no matter what the value
of y, immediately acquires values different from zero. Thus the
Fig. 172.
vorticity, originally confined within the infinitely thin sheet of dis-
continuity, is instantaneously distributed throughout the liquid, as
shown in Fig. 172, for the times -^9 — > —> 1, 4 times — Thus we
see how discontinuities of the sort shown in Fig. 166 are impossible
in nature, being replaced by the formation of eddies.
NOTES,
NOTE I.
DIFFERENTIAL EQUATIONS.
The differential equations of mechanics are of the type known as
ordinary, as opposed to partial, that is they involve a number of functions
of a single variable, the time, and the derivatives of these functions with
respect to that variable. Suppose for simplicity that we have three func-
tions #, «/, 0 of the variable ty and that instead of being given explicitly,
they are defined by the equations
If we now differentiate these equations, bearing in mind that #, «/, 0 are
dependent on t, we obtain
dF, dx dF, dy dF, dz dF, _
dt "1" dy dt "*" ds dt 1' dt ~
dF, dx dj\ dy dF, dz dj\ _
3x dt "t" dy dt "l" dz dt "*" dt ~ '
dj\d^ dj\dy_ dj\d^ dj\ =
dx dt "•" dy dt ~* dz dt ^ dt
Suppose now that the functions F contained, besides the variables indi-
cated, certain constants, c1? C2 . . . Each time that we obtain an equation
by differentiation, we may utilize it in order to eliminate from the
equations 1) one of the constants c. Thus we obtain (since the partial
derivatives are given functions of x, y, z, f), instead of the equations 1),
the following,
3)
which, since they contain the derivatives -^i -j^i -^t are differential equa-
tions, of which equations l) are said to be integrals.
If we again differentiate equations 2), we obtain
, dF, d*x d*F, /day d*Ft dxdy _
~fa ~w + ~d^ (~dt) + 2 --^
560 NOTES.
which we may again use to eliminate constants c from 3), so that
instead of l) or 3) we now have the system
. dx dy dz d*x d*y d* z\
y, M, -, -> , f , = 0,
~)
=0,
These differential equations, since the order of the derivatives of the
highest order contained in them is the second, are said to be of the
second order. In like manner we may continue, and successively eliminate
all the constants cx, C2 . . ., obtaining differential equations of successively
higher orders. Reversing the process, each set of a given order is said
to be the integral of the set of order next higher.
Any of the sets of differential equations represents the functions
x, y, #, but with the following distinction. If the equations 1) contain
constants, to which different values may be assigned,
6) Ft(x, y, 0, t, CJL, c2 . . . c^ = 0, F9(x, y, z, t, c1? c2 . . . cn) = 0,
.F30,2/, *, t, ct, c2 . . . cB) = 0,
for every set of values that may be assigned to the constants, a different
set of functions is represented, so that we have an infinity of different
functions, the order of the infinity being the number of constants contained
in the equations. Now the differential equations obtained by eliminating
the arbitrary constants represent all the functions obtained by giving the
constants any set of values whatever. Thus the information contained
in the differential equations is in a sense more general than that contained
in the equations 6), in which we give the constants any particular values.
If we reverse the process which we have here followed to form the
differential equations, we see that every time that we succeed, by 'inte-
gration, in making derivatives of a certain order disappear, we introduce
at the same time a number of arbitrary constants equal to the number
of derivatives which disappear. Thus the integral equations of a set of
differential equations of any order will contain a number of arbitrary
constants equal to the order of the differential equations multiplied by
the number of dependent variables. As an example consider the very
simple case of equations 38), § 13.
38")
d ~ » - ' dt* ~ '
Integrating these we obtain
39) x = c±t + d±1 y = c2t+d2, z = c3t + d3,
containing the six arbitrary constants c,, c2, c3, d1? d%, c?3. The meaning
of these integral equations is that the point x, y, z describes a straight
line with a constant velocity. But the differential equations 38) represent
I. DIFFERENTIAL EQUATIONS. 561
the motion of a point describing any line in space with any velocity.
Now there are a four -fold infinity of lines in space, and a single
infinity of velocities. We therefore see the very general nature of the
information contained in the differential equations. So in the example
of § 13 the statement that all the planets experience an acceleration
toward the sun which is proportional to the inverse square of the
distance expresses a very general and simple truth, in the form of a set
of differential equations, while the integral states that the planets describe
some conic section in some plane through the sun, in some periodic
time, all the particulars of which statement are arbitrary.
The characteristic property of the differential equations of mechanics,
for the phenomena furnished us by Nature, is apparently that they are
of the second order. This , although leaving possibilities of great generality,
suffices to limit natural phenomena to a certain class, in contrast to
what would be conceivable. For the consequences of the removal of
this limitation, the student is referred to the very interesting work by
Konigsberger, Die Principien der Mechanic.
In order to determine the particular values of the arbitrary constants
applicable to any particular problem, some data must be given in addition
to the differential equations. It is customary to furnish these by stating
for a particular instant of time, the values of the coordinates of each
point of the system, and of their first time -derivatives, which amounts
to specifying for each point its position and its vector velocity for the
particular instant in question. This furnishes six data for each independent
point, which is just sufficient to determine the constants. Thus if we
are dealing with a system of n points free to move in any manner,
under the action of any forces, the statement of the problem will consist
in the giving of the differential equations
dt ' dt ' dt*' d
together with the so - called initial conditions , that for t = tQ ,
% = V, 2/i = */i° - • • • zn = %n
dxi _ iy no fyi _ r.J no . . ^ _ rJ 10
~dt ~ I*1-1 >~di ~ Ll/lJ ' ~dt " L*wJ '
From these it is required to find the integrals
% = /i(0» 2/i = /i(0» *i = /s(0» • • • *« = fi-W-
Cases involving the motion of points whose freedom of motion is limited
are dealt with in subsequent chapters.
WEBSTER, Dynamics. 36
562 NOTES.
NOTE II.
ALGEBRA OF INDETERMINATE MULTIPLIERS.
On page 61 we have an example of the use of indeterminate
multipliers in elimination. It may be somewhat more clear if we examine
in just what the process involved consists Equation 12) is a linear
equation involving the 3n quantities d^, ... dzn, each multiplied by a
coefficient which is independent of the d's. Besides this equation the
quantities d satisfy the equations 14), which are of the same form, that
is, linear in all the $'s, with coefficients independent of them. Aside
from this the d's may have any values whatever. It is for the purposes
of this discussion quite immaterial that the d's are small quantities, we
are concerned simply with a question of elimination. Let us accordingly
represent them by the letters x^ X2, ... #m, between which we have the
linear equation
1) A^ 4- A2x2 4- ---- h Amxm = 0.
The x's are however not independent, but must in addition satisfy the
equations
B^XI 4- B12x2 - - - 4- BimXm = 0,
B^ 4- #22^2 ---- H
• 4-
The number of these equations, &, is less than »z, the number of the x's.
The question is now, what relations are involved among the A's and J5's
when the x's have any values whatever compatible with the equations 2).
We may evidently proceed as follows. Transposing m — k terms
in 2), say the last, we may solve the equations for the quantities
#1? X2 • • • xki as linear functions of the remaining #A_J_I, . . . xm. These
m — k quantities are now perfectly arbitrary. Inserting the values of
x± . . . Xk in equation l), this becomes linear in the m — k quantities
afc+i, . . • xm, which being purely arbitrary, in order for equation l) to
hold for all values of the x's, the coefficient of each must vanish, giving
us the required m — k relations between the A's and B's.
Instead of proceeding in the manner described, the method of
Lagrange is to multiply the equations 2) respectively by multipliers
Aj, yl2, . . A*, to which any convenient value may be given, and then
to add them to equation l). We thus obtain
(Al 4- *! -Bn 4- A2 -Bai • • • 4-
-f (A2 + A! 512 + A2 £22 • • • 4-
4- (4»4- *i-Blm+ AS •»»»»• ' ' 4-
III. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 563
In this equation the afs are not all arbitrary, but as before & may be
determined as linear functions of the remainder, say a?1, . . xk in terms
of #£_f.i, . . . flJ/n, which are arbitrary. But the multipliers A are as yet
arbitrary. Let us determine them so that they satisfy the equations
A + *i #11 + *2 #21 ' ' • + A* JBti = 0,
v ^-2 ~l~ ^1 #12 ~
#22 * ' * -f ^/fc#&2 = 0,
J-yfc -f ^i #1 A + Aj #2* ' ' * ~
which are just sufficient to determine them We thus have
+ At -Bl, yfc-fl + • • • +
5)
+ (Am 4 A! 5lm + • - - + A4 Skm)xm = 0,
in which the x's are all arbitrary, so that the m — k coefficients must
vanish, giving the m — If equations.
= 0
=0.
Inserting in these the values of the I's already found, we have the
m — Jc required relations between the A' 8 and J5's. Obviously the result
of the elimination may be expressed in the form obtained by writing
equal to zero each of the determinants of order fc + 1 obtained from the
array of A's and J3's in equations 4) and 5) by omitting m — ft — 1
rows, only m — & of the determinants thus obtained being independent.
NOTE III.
QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS.
The method of transforming the equations of motion used in § 37
and the application of hyperspace there occurring render a somewhat
more detailed treatment of the question desirable. In order to elucidate
matters, we will begin with the very simple case of a space which is
included in ordinary space, namely the space of two dimensions forming
the surface characterized by two coordinates ^n #»» as on Pa£e HO-
We have seen that this space is completely characterized by the expression
for the arc as the quadratic differential form
A point lying on this surface may be displaced in any manner, in or
out of the surface. If it is displaced in the surface, its displacement is
a vector belonging to the two-dimensional space considered. We will
36*
564 NOTES.
call the changes dql,dq2 the coordinates of the displacement. We have
found that when the displacement is made so as to change only one of the
coordinates of the point ql or q2. the arcs are respectively ds^ = y'Edql,
ds% = yrGrdq2, and that the angle included by them is given by
F
cos # =
If now we have any displacement ds, whose coordinates are dq1, dq2,
and project it orthogonally upon the directions of ds1, ds2, we easily see
(Fig. 26) that the projections daL, d62 are
*„, = ds, + ^ cos & =
da, = ds, + ds, cos 4> = ya dq2 +
We shall now, following Hertz, introduce the reduced component of the
displacement along either coordinate -line, defined as the orthogoneal pro-
jection divided by the rate of change of the coordinate with respect to
the distance traveled in its own direction. These reduced components
we shall denote by a bar, so that
3)
The fundamental property of these reduced components is found in the
equation giving the magnitude of the displacement
4) ds2 = dq^dq^ + dq2dq2,
that is the square of an infinitesimal displacement is the sum of products
of each coordinate of the displacement multiplied by the respective
reduced component.
In like manner the geometric product of two different displacements
ds, ds', whose coordinates are dq^, dq2, dq±\ dq% is found to be
' cos (ds, ds') = dxdx' -f dydy' + dzdz'
dx , dx -. \ fdx , , , dx
(
6)
q2'-l-dq2dql') + Gdq2dq2'
dq2dq2' = dq^dq^ + dq2'dq2.
HI. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 565
The geometric product of two displacements is equal to the sum of
products of the coordinates of either vector by the reduced components
of the other. Thus the geometric product is denned by means of the
quadratic differential form l) denning the space in question.
Solving the equations 3) for dqv dq%, we obtain
6)
from which we obtain
7) ds2 = En dq^ -f 2 J212 dq± dq2 -f E22 dq22.
The expression 7) is called the reciprocal form to l). Corresponding to
it we obtain the form of the geometric product
We may now define any vector belonging to the space considered,
as one whose components have the same properties as those possessed
by those of an infinitesimal displacement. Suppose that X, Y, Z are the
rectangular components of a vector E, it does not belong to the space l)
unless it is tangent to the surface in question. If so, we have a displace-
ment such that
ds dx dy dz dql dq^ s
Then Q^ Q2 are the coordinates of the vector in the system gA, 2, and
the magnitude of the vector is given by the equations
E2 = X2 + Y2 + Z2 = ~(dx2 -f dy2 +
10) = (Ed^i2 + 2Fdqidq2 -f
where
Ci
%
are its reduced components belonging to its coordinates Q^ Q%. The
geometric product of two vectors J2, It' is
12) «i«if+«8«a'-«i'«i+«8'«a-
If now one of the vectors is finite, the other an infinitesimal displacement,
we have the geometric product
566 NOTES.
Thus the reduced components ft, Q2 are given by the definition which
we have adopted on page 116, equation 42).
Having now illustrated the subject by a space of two dimensions,
we can easily extend our notions to space of any number of dimensions
m, defined by the form
14) ds*
For any one of the coordinate directions we have
15) dsr* = Qrrdqr\
and for the geometric product of two displacements,
16) dsds1 cos (ds, ds') = ^ ^s Qrsdqrdqs'.
If one is in the direction dsr, all dq's being 0 except dqr,
17) dsdsr cos (ds, dsr) = ^.s Qrsdqrdqs,
and dividing by dsr we obtain the orthogonal projection of ds on dsr
18) d6r = ds cos (ds, ds^ = ^_ g%
VQrrd(lr
from which we obtain by the definition of the reduced component
da
19)
dr
We have as before
20)
21) dsds' cos (ds, ds1)
and if the solution of 19) is
22)
we have the reciprocal form
23) ds*
Again we may define a vector belonging to the hyperspace considered,
and now the rectangular components may be of any number, the limitation
of the vector to the space in question reducing the number of generalized
IV. AXES OF CENTRAL QUADRIC. 567
components to accord with the number of dimensions of the hyperspace.
The geometric product of the vector with an infinitesimal displacement
defines the generalized coordinates of the vector, so that
Yr dljr + Zr
r=n «=
and we find that the reduced component of the vector is what is defined
by the formula 42) of page 116)
In our application to mechanics the differential form in question is
2Tdt2, where T is the kinetic energy of the system It is immaterial
whether we speak of vectors in a hyperspace, as we have here done, or.
as Hertz does, speak of vectors with respect to our mechanical system.
The meaning in either case is plain. On dividing the above formulae
by dt2, we find that the generalized velocities and momenta have the
relation to each other of coordinates and reduced components of the same
vector in the hyperspace. The two reciprocal forms 14) and 23) have
the relation of the Lagrangian and Hamiltonian forms of the kinetic
energy. The equations of motion of the system say that, no matter
how the forces are applied, or how parts of them are equilibrated by
the constraints, the reduced components of the applied and the effective
forces are equal for every coordinate.
NOTE IV,
AXES OF CENTRAL QUADRIC.
The principal axes of a central quadric surface,
1) F(x, y, e) ~ Ax2 + By* + Cz2 -f %Vyz + 2Egx + 2Fxy = 1,
are defined as the radii vectores in the directions for which the radius
vector is a maximum or minimum. If we put
x = ra, y = r|3, z = ry,
we have
2) ^ = F(«, P, y),
and the maxima and minima of r occurring for the same directions as
the minima and maxima of 1/r2, are obtained by finding the maxima and
minima of F(cc, ft, y) subject to the condition
3) g>(a,
568
NOTES.
If we multiply this equation by an arbitrary constant — /I and add it to
_F(a, |3, y), we obtain the condition by writing the derivatives of F — lq>
equal to zero Thus we obtain
4)
Now the direction cosines of the normal to the quadric at a point #, y, &
are proportional to
^E. dJL dJL.
dx dy dz
At points where the normal is in the direction of the radius vector we
have
dF dF dF
dx _ dy dz
x y z
But
dF(x, y, z) dF(cc, ft y)
dx
da
etc.,
so that the equations 4) show that at the ends of the principal axes the
tangent plane is perpendicular to the radius vector.
Effecting the differentiations the equations 4) become
(A -
5)
F0
Dfi
7 =0,
y =0,
-h(<7-/l)y=0.
The condition that these equations, linear in a, |3, y, shall be compatible
for values of a, |3, y, other than zero is that the determinant of the
coefficients shall vanish.
— I, F ,
F , B - A,
E
D
E
D
(7
This is a cubic in A, which being expanded is
IV. AXES OF CENTRAL QUADRIC. 569
We shall show that this always has three real roots. Put
A — X = u -\- q,
8) B - I = v -f r,
C — I = w + S,
where #, r, 5 are to be determined later. Then
0*
or arranged according to powers of u, v,'w,
uvw 4- qvw + rww +
9) + u(rs - D2) + v(sq - iJ2)
+ gr5 + 2DEF - D2q - E2r - F2s = 0.
Let us now determine q, r, s, so as to make the terms of first order in
Uj v, w vanish.
rs = D2, sq = E2, qr = F2,
from which by multiplication and division
10) qrs =
Thus there remains
, EF FD . DE
11J uvw -\ — =r- v w H — ^- ivu H — =- uv = 0.
Now from 8)
u = A — H — q = A — EF/D - K = a — A,
12) v = B -X-r = B - FD/E - I = b - A,
w=C -l-s = C - DE/F - I = c - A,
if we write
A - EF/D = a, B-FDjE=l, C-DE/F=c.
Also since from 10) #, r, 5 are all of the same sign, let us call them
+ Z2, w2, w2, so that we have from 11)
/•(T) - (a - A) (6 - A)(c - A) ± [Z2(fc - A) (c - A)
+ M2(c - A) (a - A) + n\a - I) (b - A)J.
Substituting for A in turn the values — oo, c, ft, a, +00, we obtain
/•(-oo)=oo,
/>(c) =±^2(«-C)(&-C)
14)
/"(a) =
/•(oo) - - oo.
570 NOTES.
Let us suppose a > b > c. and take the upper sign in 14). Then for
I = — oo f(l] is 4-
c 4-
&
o +
00
and the function f(l) behaves as shown in Fig. 13. As there are three
changes of sign, there are three real roots. It is to be noticed that the
reality of all the roots depends on #, r, s being of the same sign. Let
us call the roots A15 A2, A3. Either one of these being inserted in the
equations 5), the equations become compatible, and suffice to determine
the ratios of the direction cosines. There are therefore always three
principal axes to a central quadric surface. If we call the cosines
belonging to the roots Ax, a1? /31? y1? those belonging to A2, «2, /32, y2,
equations 5) become
15)
Multiplying the first three respectively by a2, /32, yt and adding,
4-
,
4- D(ftyi 4- ft72 4- ^y2^ + yi«2 4-
If we multiply the second three equations respectively by a1? jSj, 7l and
add we obtain for
the same expression. Accordingly we have
17) (^ - A2) (X a, 4- ft ft + 7l 72) = 0,
so that if the roots /Lj, ^2 are unequal the corresponding axes are
perpendicular. In like manner if the determinantal cubic has three unequal
roots, the quadric has three mutually perpendicular principal axes.
If two roots are equal the position of the corresponding axes becomes
indeterminate, and it may be shown that all radii perpendicular to the
direction given by the third root are principal axes of the same length.
The surface is then one of revolution about the determinate axis. If all
three roots are equal, the surface is a sphere, and any axis is a
principal axis.
IV. AXES OF CENTRAL QUADRIC.
571
We will now transform the equation of the quadric l) to a new set
of axes coinciding in direction with its principal axes. Let the new
coordinates be x\ y\ #', and let the direction cosines of the angles made
by the new with the old axes be given in the table below.
The equations of transformation of coordinates are then
x = a^x + ft?/ -f yis,
18) y1 --= a^x -f ft?/ -f 722,
x = a^x' 4-
19) y = (llX' +
' -f
Now using equations 19), we obtain
ft
which in virtue of equations 15) is equal to
*!«!#' + ^22/'
In like manner
Multiplying respectively by #, «/, ^ and adding, we obtain
Ax2 + By* + £2 + 2D«/^ + 2Ezx
pi l^x'fax + ft?/ + 7^) + ^fax -f ft«/ + 72*) +
Consequently the equation of the quadric referred to its principal axes is
20) AX2+ V2-fM'2=1,
and the three roots of the cubic are equal to the squares of the reciprocals
of the lengths of the semi -axes. Accordingly in order to find the equation
referred to the axes it is not necessary to solve the linear equations 5),
but only to solve the cubic 6).
572
NOTES.
NOTE V.
TRANSFORMATION OF QUADRATIC FORMS.
The last two notes have dealt with quadratic forms, and in Note IV
we have by a linear transformation of the variables 19) transformed
the form F into a form 20) in which no product terms appear, and we
find that the coefficients of the squares are the roots of the determinant 6).
In this note we shall consider similar transformations of forms of any
number of variables, and shall incidentally obtain a proof of the reality
of the roots of Lagrange's determinant, 65), page 159, for the case of
no dissipation.
We shall require a number of elementary properties of both linear*
and quadratic forms, which we shall now set forth. Suppose we have n
linear forms
1)
a2nccn,
and let us call jR the determinant
Clio.
E =
"11 »
If we multiply the &th column of E by ##, and then add to this column
the first, second, etc., multiplied respectively by #1T #2,
Ol,*-f li • • • Oi?
we obtain
3)
#1,* — 1,
ann
If now the determinant R is zero, the determinant on the right vanishesT
expanding which we obtain
4) c^ -h c%u2 + h cnun = 0,
where the c's are the minors of the elements of the fcth column of E.
Thus if the determinant of the forms vanishes, the forms are not independent,
but satisfy identically the linear relation 4).
Consider now the quadratic form
r = n s=n
V. TRANSFORMATION OF QUADRATIC FORMS. 573
for which ars = asr, and
K 2/i5 2/2? • • • 2/w are another set of variables, and we put for each xr
the value xr + lyr we have
ars (xr -f- ^«/r)(^s
r = l * = 1
r = n s=n r = n s=-n
8)
If now -R, the determinant of the form f vanishes, we have a relation
r = n
9) c^ + c2^2 H ----- h cwwn EE 'fyfffa, . . . a?w) = 0
for «W values of #1? ... a;fl.
Let us now put for the y's of equation 8) the values of c of
equation 9). We then have by 7) and 9)
^ri . - • O = 0,
r=l
so that 8) becomes
10) +
We thus find that in this case f is independent of I and of yl , . . . yn.
xn
Accordingly if cn is not zero, let us put I = ? so that we obtain
Cn
for all values of xlt . . . xn,
C C Cn — l
11 J / (j£j, . . . Xn) = t(X^ — Xn, #2 — Xn, . . . Xn — i - /£ Q\
c« ^n ^n
Thus we obtain the theorem: every quadratic form in n variables
whose determinant vanishes, may be expressed as a quadratic form of
less than n other variables «/, which are linear combinations of the original
574
NOTES.
variables. Such forms are called singular, as opposed to ordinary forms,
whose determinant does not vanish. As an example the form
^2 * ^3 '
111 0
1 1 0
0 0-1
whose determinant
vanishes, may be written
If f is an ordinary form and arr is not zero, we may write
12) f=arrxr*+2xrp + q,
where p is a linear form containing all the other variables except xr
and q is a quadratic form in the same variables. Completing the square
we may then write
13) arr f = (arr xr -f p)2 + arr g — p2.
If on the other hand every coefficient of a square arr is zero, we may write
14) f= 2arsxrxs -f 2xrp -f 2#g(? + r,
where J9, # are linear, r a quadratic form not containing either xr or #„
we may then write
2 «„/"= 4 (a^ov
-f 2 arA.
In the former case we have exhibited f as the sum of a square and a
form in w — 1 variables, in the latter as two terms in squares and a
form in n — 2 variables. In either case the remaining form may be
treated in a similar manner, and so on, so that the form is eventually
reduced to a sum of terms in squares. If the coefficients A of all the
squares in
16) f
when the y's are linear forms in xv . . . #„, have the same sign, the
form is said to be definite, for it can not change sign however the values
of the variables be altered. If the coefficients A are not all of the same
sign the form is indefinite.
If we transform the linear forms l) by means of the linear sub-
stitution
V. TRANSFORMATION OF QUADRATIC FORMS. 575
we obtain linear forms in the new variables y^ . . . yn, so that if we write
2/1 + C12 2/2
18)
= C21 2/1
H
we find by carrying out. the transformation,
19) crs--
But this is, according to the rule for multiplication of two determinants
the condition that the determinant of the forms u in y,
20)
is the product of the determinant of the forms in x by the determinant
of the substitution 17).
The determinant in which the element in the rth row and sth column
dur
is a derivative •* — is called the Jacobian of the functions u< , . . . un with
dxs
respect to the variables #19 . . . #n, and is often denoted by
Cn, . . . G\n
«m •;•••! «in
fti, • • • §ln
•=
£"n\i ' • - Cnn
Q>n\i - - - Q>nn
Pnlj ' • • Pnn
?(«,....*.)
In this notation 20) becomes
/ ~WT^i ^~\ =
fti. • • •
If the functions Mlt . . . ttn in a Jacobian are the partial derivatives
o /»
of the same function /", wr = -jr^-* so that the element in the rth row
gsr ^^r
and sth column is -~ — 5 — the determinant is called the Hessian of the
dxrdxs
function. Thus the determinant H of 2) is the Hessian of /", and will
be denoted by Hxf.
If now we transform the quadratic form 5) by the substitution 17),
so that
22)
576
NOTES.
we may find a relation between the Hessians of f with respect to the
x's and that with respect to the y's. Using the notation for Jacobians,
by 21),
23)
• • • fi
But in every derivative
8V
ay
Consequently the Jacobian on the right of 23) is the same as that on
the left of 21).
Thus we find
3(u u] fci---Pi» 2 > for which
does not vanish,
since icn is independent of these variables, we may by suitably choosing
the value of xn make the form have either sign, it is therefore indefinite.
V. TRANSFORMATION OF QUADRATIC FORMS.
577
(If the sum ^J
r=l *=1
must have
arscryis is zero for all values of y1? . . . yn-\, we
for 5=1, 2, , ... n — 1, but since f(c^ . . . cn) is zero we must have
r=n
also ^ arncr, and these n equations require the determinant of the form
r=l
to vanish, and the form is singular.)
As a result of this theorem we see that ifji form is to be definite,
no coefficient arr of a square xr2 must be absent, and all must have the
same sign*. For if arr = 0, putting all the variables equal to zero
except xr would make the form vanish, and if arr is not zero, the same
assumption would make the form have the sign of arr. Consequently all
these coefficients must be of the same sign.
Let us now consider two ordinary quadratic forms of the same
variables, with real coefficients1)
r=l s = l r = l s=l
from which with an arbitrary multiplier A we construct the form
28) Ay -f i/;.
As we give A an infinite set of real values, we obtain an infinite sheaf
of forms. Let us examine whether they are definite or not.
The determinant of the form Ago -f i/;
28)
-f-
-f
-f
-f
cnn
= f«
is identical with Lagrange's determinant, page 159, when the %'& are zero.
(We here have written I for the A2 on p. 159.) We shall now prove
that if the equation f(k) = 0 has a complex root, all the forms of the
sheaf Ago -f- i/; are indefinite.
Let A = a -f if be a complex root of the determinantal equation
/"(A) = 0. Then since the form (a 4- if) y -f ^ is singular, it may be
represented as a sum of less than n squares, and since it is complex,
these may be squares of complex variables, so that we have
29) (a
1) Kronecker, Uber Schaaren quadratischer Formen. Monatsber. der Konigl.
PreuB. Akad. d. Wiss. zu Berlin, 1868. pp. 339 — 346. Werke, Bd. I, p. 165.
WEBSTER, Dynamics.
37
578 NOTES.
the «/'s and g'a being real linear forms in the #'s, any of which, but
not all, may be zero. Separating the real and the imaginary terms.
we obtain
r==n — \ r = n — 1
30) «9 + ^
From the two forms
the whole sheaf may be obtained. Solving 30) for g> and
32) A cp + y
which we may write
r=^~ 1 / ! v
33) I Cp -f ib = >Yyr — ^#r) [9r H #r)»
^_ \ f* ./
r=l
where
^ ^ 0
a quadratic in ft, giving for every real value of A, a real value of jii.
Now each of the forms 33) vanishes for values of #15. . . xn other than
zero, which satisfy the n — 1 linear equations
and is accordingly indefinite. Conversely if there is in the sheaf a single
definite form, the roots of /"(A) = 0 are all real. Now in the mechanical
application, the form go, which is proportional to that given by the value
A = oo, is the kinetic energy, a definite positive form, consequently the
reality of the roots is proved.
If Ax is one of the real roots of the equation /"(A) = 0 the form
A*9> + ty being singular, can be expressed in terms of less than n linear
functions of a^, ...#„, say y^...yn—\. Let yn be any other linear
function of the #'s, such that the determinant of the functions «/1? . . . yn
is not zero, then we can express the function , a
definite positive form.
V. TRANSFORMATION OF QUADRATIC FORMS. 579
A linear divisor such as A — Kr of the determinant of the form
A g/ -|- tyr is also a divisor of the determinant of A g> -f- i/;, for on writing
out the determinant of the form 35) in terms of y^ . . . yn^\tsy.^ we find
36) a^ to . . . ,,„_,(* ?> + *) = (x - i.) tf* . . . r._,(V + *0,
so that the vanishing of the determinant of order n — 1 on the right
makes the determinant on the left vanish. But this equal to the deter-
minant of A (p -f- ty in the variables x± , . . . xn multiplied by a constant.
We may now treat the form hep' -\- i^f in the same manner, and so
on, so that finally we obtain
37) lg> + y = (I - Aj^2 + (I - A2>22 + - - - + (A- Aw)^2,
where A15 . . . Kn are the roots of the determinantal equation f(k) = 0.
Since this is true for all values of A we have
which is the simultaneous transformation of two quadratic forms required
in the treatment of principal coordinates. It is obvious according to
this method that it makes no difference whether the determinant has
equal roots or not.
Absolute system, 31.
Acceleration, definition of,
13.
angular, 248
centripetal, 14.
components of, 13,14.
compound centripe-
tal, 319.
constant, 33.
of Coriolis, 320.
— moment of, 17.
normal, 17.
radial, 16.
tangential, 14.
transverse, 16.
Actio, 66.
Action, definition of, 101.
least, 101.
examples of, 140, 141.
surface of equal, 139.
varying, 131.
Activity, equation of, 66.
in Lagrange's coor-
dinates, 125.
— in hydrodynamics,
501.
Addition of vectors, 5.
Adiabatic motion, 189.
d'Alembert's Principle, 63.
— statement in
words, 65.
— in generalized
coordinates, 118.
Amplitude, 35.
Analytic function, 523.
Angle, solid, 351.
Angular acceleration, 248.
— velocity of moving
axes, 246.
Anisotropic body, 457.
Aplatissement , 407.
Appell, 54, 61, 309, 316.
Archimedes' principle, 472.
Areas, law of, 18.
Atmosphere, height of, 465.
Attracting forces, 76.
Atwood's machine, 23.
Axis, central, 209.
fixed, body moving
about, 250.
INDEX.
(The numbers refer to pages.)
Axis, of suspension and
oscillation, 251.
Axes, moving, motion with
respect to, 316.
rotating, 317.
parallel, 201.
— theorem of in-
ertia, 229.
of spherical harmo-
nic, 396.
— — strain, 433.
Axioms, physical, 20.
Bacharach, 378.
Balance, 33.
Ball, 224.
Ballistic pendulum , gal-
vanometer, electro-
meter, 72.
Base-, golf, tennis ball, 34.
Basset, 499.
Beats, 156.
Beads, string of, vibration
of, 164.
frequencies of,
167.
Beltrami, 113.
Bending moment, 490.
Bernoulli, 58, 84, 173.
Bernoulli-Euler theory, 485.
Bernoulli's theorem, 505.
Bessel, 407.
Billiard ball, motion of, 305.
Binet, 231, 239.
Bodies, three, problem of, 31.
Bourlet, 309.
Boussinesq, 309, 346.
Boyle's Law, 544.
Boys, 30, 362.
Brachistrochrone, 77, 82.
Brahe, Tycho, 18.
British Association, 31.
Buckling of sections, 485.
Bulk-modulus of elasticity,
461.
Bullets, toy, 23.
Bunsen pump, 504.
Cadmium, wave-length of,
27.
Calculus of variations, 77.
Cambridge, 269.
Carvallo, 309, 313.
Cauchy, 451, 456, 459.
Cavendish, 30.
— Laboratory, 269.
Center of mass, motion of, 89.
Central axis, 209.
forces, 38.
Centripetal acceleration ,
compound, 319.
Centrifugal force, 119.
— — on earth, 320.
— in rigid body,
228.
Centripetal acceleration, 15.
— — compound, 319.
Centrobaric body, 364, 404.
C. G. S. system, 33.
Chasles, 212, 213.
Chasles's theorem, 373, 413.
Characteristic function, 136.
— of plane, 214.
Circle, Dirichlet's problem
for, 388.
Circular Harmonic, 388.
— — development in.
390.
Circulation, 506.
Clairaut, 33.
Clairaut's theorem , 404,
407.
Clebsch, 224, 478.
Clifford, 87, 521.
Clark University, 268.
Coefficients of inertia, stiff-
ness, resistance, 158.
Coffin, 52.
Complex, 215.
— equation in line coor-
dinates, 221, 223.
— variable, 521.
Component, of vector, 116,
— — generalized,! 16,
— of momentum, 117,
Composition of screws, 216
Concealed motions, 179.
Concentration, 345.
— proportional to den-
sity, 360.
INDEX.
581
Condition for equipotential
family, 410.
Cone, rolling, 253.
of equal elongation,
443.
Confocal quadrics, 235.
Conjugate functions, 523.
Conical pendulum, 55.
Conic section orbit of pla-
net, 18.
Conservation of motion of
center of mass, 90.
Connectivity of space, 339.
Conservation of energy, 68.
— — integral of La-
grange's equations,
126.
Conservative system, 65, 68.
Constant of gravitation, 30.
Constants of elasticity, phy-
sical meaning of, 461.
Constraint, equations of, 57.
non-integrable,
equations of, 313.
varying, 129.
Continuity, equation of, 498.
Convective equilibrium, 466.
Coordinates, 5.
curvilinear, 110, 330.
cyclic, 176.
cylindrical, 335.
ellipsoidal, 234, 335.
generalized, 109.
line, 215.
normal, 163.
orthogonal, 110, 330.
polar, 11.
positional, 189.
principal, 163.
ignoration of, 179.
Coriolis, 317.
— theorem of, 319.
Corner, flow around, 528.
Cornu, 491.
Correction for finite arcs, 48.
Cotes1 s spiral, 519.
Coulomb, law of.
Couple, 204.
arm, 205.
causing precession,
299.
— composition of, 208.
— in regular preces-
sion, 274.
Couple, of forces, 205.
— moment of, 205.
— righting, 473.
— theorems on, 207.
Curl, 87.
— in curvilinear coor-
dinates, 383.
Curvature, 16.
Curve, expression for arc
of, 111, 113.
— parametric represen-
tation of, 10.
— tautochrone, 144.
Curvilinear coordinates,
Green's theorem in
380.
Cusps on curves of equal
action, 140, 141.
Cyclic coordinates, 176.
— systems, 188.
— — examples of,193.
reciprocal rela-
tions in, 191.
— — work done on,
192.
Cycloid, 84.
as tautochrone, 145.
— drawn by point of
top, 287.
Cycloidal pendulum, 148.
Cylinder under pressure,
475.
— moment of inertia
of, 242.
Cylindrical coordinates,335.
flow, 653.
Cylindroid, 218.
Damped oscillations, 148.
Damping, coefficient of, 151.
Darboux, 113.
Decrement, logarithmic,
151.
Deformable bodies , kine-
matics of, 427.
— — statics of, 463.
Density, 353.
Derivative, directional, 88,
331.
— of analytic function,
523.
— particle, 497.
Development in circular
harmonics, 390.
Development of reciprocal
distance, 398.
— in spherical harmo-
nics, 400.
— of potential in har-
monics, 402.
— of potential of ellip-
soid of revolution,
424.
Differential, perfect, 88.
— equation for forced
vibration, 152.
— — of Legendre,
398.
— — of particle un-
der Newtonian law,
39.
— parameter, 88, 330.
— — arithmetical va-
lue of, 333.
invariant, 333.
— — mixed, 343.
— — second, 344.
Dilatation, 436.
Dimensions of units, 27,
28, 29.
Dimensional, two, poten-
- tial, 385.
Dirichlet, 69, 376, 377, 378,
393, 402, 417.
Dirichlet's problem, 376.
— for circle, 388.
— — for sphere, 395
Disc, moment of inertia
of, 242.
Displacement, infinitesimal,
59.
virtual, 58.
— lines of, 446.
Distributions , energy of,
425.
— surface, 367.
Dissipation, 122.
— function, 123.
Divergence, 347.
— theorem, 347.
Double-lines , complex of,
214.
Driving points, 123.
Dualism, 209.
Dyne, 29.
Earth, motion relative to,
320.
582
INDEX.
Echo, 545.
Ellipse , equation relative
to focus, 18.
Ellipsoid of elasticity, 451.
of gyration, 233.
— inverse, 233.
— Jacobi's, 470.
— Maclaurin's, 469.
— moment of inertia
of, 241.
rolling, in Poinsot-
motion, 259.
potential of, 415.
— — at internal
point, 418.
differentiation,
419.
of revolution, poten-
tial of, 421.
Ellipsoidal coordinates, 234,
335.
— as equipotential
family, 412.
Elliptic function, 45.
— integral, 45, 47.
Ellipticity, 407.
Elongation and compres-
sion quadric, 441.
Energy, 65.
— conservation of, 68.
— equation of, 67.
— of distributions, 425.
in terms of field, 426.
— emitted by sound-
source, 548.
— function for isotro-
pic bodies, 457.
— invariant, 457.
integral of, in top,
277.
— kinetic, 66.
— general form of,
112.
maximum or mini-
mum for equilibri-
um, 69.
potential, 68.
— — exhaustion of,76.
— all due to mo-
tion, 182.
— of normal vibrations,
164.
— relative kinetic, 92.
— of vortex, 515.
Eolotropic body, 457.
Epicycloid, on polhode, 270.
Epitrochoids described by
heavy top, 293.
Equation of activity, 66.
— — in hydrodyna-
mics, 501.
— — in generalized
coordinates, 125.
— of continuity, 498.
— differential, of mo-
tion, 24.
— — of forced vibra-
tions, 152.
— of equilibrium for de-
formable body, 448.
— of hydrodynamics by
Hamilton's principle,
500.
— Euler's, for rotation,
261.
— Laplace's, 349.
— of motion from Ha-
milton's principle,
114.
— — from least ac-
tion, 106.
— — Lagrange's first
form.
— — — second
form, 115.
— — Hamilton's ca-
nonical, 128.
— Poisson's.
— — applied to earth,
361.
— — for two dimen-
sions, 387.
Equilibrium, 25.
— conditions for, 61,69.
— stable and unstable,
69.
— convective, 466.
— equations of for de-
formable body, 448.
— theory, 155.
of tides, 535.
Equipollent loads, 486.
Equipotential surfaces, 354.
— — condition for fa-
mily of, 410.
— layers, 372.
— surface of strain, 446 .
Erg, 56.
Erg, unit of energy, 70.
Ether, luminiferous , 65.
Euler, 73, 173, 499.
— equations of hydro-
dynamics, 499.
— angles of, 274.
— dynamical equations
of, 260.
— kinematical equa-
tions of, 275.
— theorem of 114, 127.
Everett, 29, 33.
Existence -theorem, 378.
Experiment, comparison
with theory, 51, 52.
Expansion -ratio, 434.
Falling body affected by
earth's motion, 323.
Faraday, 363.
Field, energy in terms of,
426.
— strength of, 353.
Fixed point, motion about
252.
Fleuriais, 296. ,,
Flexion, 489.
Flexure, uniform, 490.
— non- uniform, 494.
Floating body, equilibrium
of, 471.
Flow around corner, 528.
Fluid, perfect, 458.
— rotating, 467.
— gravitating rotating,
468.
Flux of vector, 349.
Flux -function, 516.
Focus, kinetic, 105.
— of plane in null-
system, 212.
Formula, Green's, 370.
— — for logarithmic
potential, 388.
Force, definition of, 24.
— accelerational, 120.
— central, 38.
— centrifugal, 119.
— — in rigid body,
228.
— component, genera-
lized, 114.
— effective, 64.
INDEX.
583
Force, effective, generalized
component of, 118.
function, 68.
— — particular case
of, 73.
— — for Newtonian
law, 75.
containing time,
69.
— of inertia, 64.
— gyroscopic, 185, 278.
— momental, 120.
motional, 26.
— n on -momental, 119.
— parallel, 205.
— positional, 26.
— reduction of groups
of, 299.
tidal, 408.
Forced vibrations, 152.
Formula of Green, 370.
Foucault, 257, 324, 325, 326.
pendulum of, 324.
gyroscope of, 324.
Fourier, 173, 393.
— coefficients in series
of, 392.
Freedom, degrees of, 58.
Free vector, 199.
Frequency, 35.
Friction, effect on top, 303.
Fujiyama, 367.
Function, analytic, 523.
characteristic, 136.
flux, 516.
— Hamiltonian, 117.
— Lagrangian, 115.
linear vector, 428.
normal, 171.
— fundamental
property of, 172.
of point, 88.
principal, 132.
self - conjugate , 430.
— of St. Venant, deter-
mination of, 483.
g, formula for, 406.
g, value of, 32, 33.
Galileo, 3, 32, 58
Gauss, 32, 312, 350, 351,
373, 374, 387, 425,
426.
— differential equation
of, 312.
Gauss, theorem of, 350, 387.
— — on energy, 425.
— — of mean, 374.
Generalized coordinates,
109, 111.
— velocities, 112.
Geodesic line, 103.
Geodesy, application of
spherical harmonics
to, 404.
Geometric product, 7, 116.
Geometrical representation
of stress, 450.
Geometry of motion, 3.
Gleitung, 441.
Glissement, 441.
Gravitating rotating fluid,
468.
Gravitation, constant of, 30.
— kinematical state-
ment of, 20.
— universal, 29.
Gravity, center of, 90.
— terrestrial, potential
of, 405.
Green, 340, 343, 370, 379,
386, 388, 456.
— formula of, 370.
— — for logarithmic
potential, 388.
— theorem of, 340.
— — in curvilinear
coordinates ,
— — for plane, 386.
Griffin mill, 273.
Gyration, ellipsoid of, 232.
Gyroscope, 274.
— in torpedo, 257.
— as compass, 326.
— latitude by, 325.
Gyroscopic forces, 185, 278.
terms, 185.
— pendulum, 282.
— horizon, 296.
— system, curves drawn
by, 293, 294.
Gyrostat, 186.
— stability of spinning,
291.
Hadamard, 313.
Hamilton, 7, 21, 97, 126,
128, 131, 176.
— equations of, 126.
Hamilton, equations from
Hamilton's principle,
127.
— method of, 136.
— principle of, 97.
— — equations of hy-
drodynamics by, 499.
— — — of string by,
170.
— partial differential
equation of,; 135.
— theorem of, 135.
Hamiltonian function, 127,
128.
Harmonics, circular, 388.
— — development in,
390.
— of pipe, 546.
— spherical, 393.
— — examples of,394.
— — forms of, 395.
— — axes of, 396.
— — in spherical
coordinates, 398.
— — zonal, 397.
— of string, 169.
Harmonic function, 345.
— ~ motions, 35.
— — elliptic, 36.
Harkness and Morley, 378.
Harnack, 388.
Hay ward, 324.
Heat, dynamical theory of,
64.
Heaviside, 87.
Height, metacentric, 474.
— of atmosphere, 404,
408, 465.
Helmert, 33, 362, 407.
Helmholtz, 3, 86, 176, 179,
458, 507, 509, 511,
512.
— energy form, 458.
— theorem of, 512.
Herpolhode, 262, 263.
Hertz, 21, 113, 182, 193.
Hilbert, 378.
Heterogeneous strain, 444.
Hodograph, 19.
Homoeoids, ellipsoidal, 409.
Homogeneous strain, 428.
Horse-chestnut, toy, 23.
Hooke, 455.
Hoop, rolling, of, 308.
584
INDEX.
Howell torpedo, 272.
Huygens, 148.
Hydrodynamics, 496.
equations of, 497.
Hydrostatics, 463.
Hyperellyptic function for
top,. 303.
Hypergeometric series, 312.
Hyperspace, 113.
Hypocycloid on polhode,
270.
Hypotrochoids described by
heavy top, 293.
Ignoration of coordinates,
179.
— example of, 181.
Impulse, 70.
— and velocity, geome-
tric product of, 72.
Impulsive forces, 71.
— — in Lagrange's
equations, 134.
Impulsive wrench, 225.
Indeterminate multipliers,
62.
Inertia, 21.
axes of, distribution
in space, 239.
coefficients of, 112.
ellipsoid of, 231.
— force of, 64.
moments of, calcu-
lation of, 241.
principal axes of,229.
— moments of, 231.
— products of, 227.
Infinitesimal arc, area, vo-
lume, 338.
Integral of function of com-
plex variable, 524.
Interferences, 156.
Invariable axis, motion of
in body, 266.
— — and plane, 95.
Invariants of strain, 457.
Invariant, second differen-
tial parameter, 347.
Irrotational motion, 520.
Isochronous vibration har-
monic, 146.
Isocyclic motion, 189.
Isotropic body, stresses in
460.
Isotropic body, energy
function for, 457.
Ivory's theorem, 420.
Jacobi, 135, 297, 380, 470.
— ellipsoid of, 470.
— method of, top equa-
tions by, 297.
Jordan, 402.
Kater, 251.
Keely, 153.
Kelvin, 3, 376, 378, 511, 538,
557.
— and Dirichlet's prin-
ciple, 376.
Kepler, 18, 38.
laws of, 31.
Kilogram, weight of, 32.
Kinematics, 3.
— of deformable bo-
dies, 427.
— of rigid system, 243.
Kinematical equations of
Euler, 276.
Kinetic energy, general
form of, 112.
— — due to rotation
249.
focus, 105.
potential, 179.
linear terms in,
184.
— reaction, 64, 65, 119,
120.
Kinetoscope, 23.
Kirchhoff, 69, 499.
— energy function of,
459.
Klein and Sommerfeld, 273.
Kneser, 105.
Korteweg, 316.
Krigar-Menzel, 362.
Kronecker, 378, 393.
to
Lag, in forced vibration, 153.
Lagrange, 155, 157, 164,
173, 507, 508.
equations of motion,
first form, 108.
equations of motion,
115.
by direct
transformation, 115.
Lagrange, equations for
small oscillations,
158.
— — of equilibrium,
62.
— — for pure rolling,
313.
— determinantal equa-
tion, 159.
— — roots of, 160.
— method of, top equa-
tions by, 279.
Lagrangian function, 115.
— modified, 179.
method in hydro-
dynamics, 499.
Lamb, 499.
Lame, 329, 331, 344, 380.
Lame's shear -cone, 451.
— stress-ellipsoid, 451.
Lamellar vectors, 87.
Laminar flow, 554.
Laplace, 3, 398, 401, 402r
542, 544.
— equation of, 349.
in spherical and
cylindrical coordina-
tes, 383
satisfied by po-
tential, 357.
— operator, 349.
Laplacian, 350.
Law of areas, 18, 38.
— Coulomb, 486.
inverse squares, 20.
Kepler, 18.
Lenz, 193.
motion, 20.
Layers, equipotential, 372.
Least action, 99.
Legendre, 47, 399.
— differential equation
of, 398.
polynomials of, 397.
Lenz's law, 193.
Level surface, 88, 89.
of potential
function, 329.
sheet, 89.
Line-coordinates, 215, 221.
Line-integral, 84.
Line -integral, independent
of path, 87.
Lines of force, 354.
INDEX.
585
Lines of vector -function,
348.
Linear terms in kinetic
energy, 130.
— — — potential,
184.
— vector function, 428.
Lines of displacement, 446.
Liquids, fundamental the-
orem for, 465.
Logarithmic decrement,
151.
— potential, 385.
Loops, 546.
Maclaurin, 415.
Maclaurin's ellipsoid, 469.
— theorem, 414.
Maupertuis, 97.
Mass, 23.
— dynamical compari-
son of, 23.
center of, 90.
Material point, 21.
Matter, 3.
— definition of, 64.
Maxwell, 123, 126, 268,
345, 349, 551.
Maxwell's theory of elec-
tricity, 124.
— top, 268.
— — effect of friction
on, 304.
Mean, integral as, 70, 90.
theorem , Gauss's,
374.
Mechanics, 3.
problem of, 26.
Mechanical powers, 60.
Mendenhall, 367.
Metacenter, 474.
Metacentric height, 474.
Metre prototype, 26.
Michell, 533.
Modulus, bulk, 461.
— shear, 462.
— Young's, 462.
Momentum, 26.
generalized compo-
nent of, 117.
Mobility, coefficients of, 126.
Mobius, 212.
Moment, axis of, 7.
— bending, 490.
, Dynamics.
Moment of inertia, 93, 227.
— of momentum, 95.
— of velocity, 12.
Momentum, 26.
— moment of, 95.
— — conservation of,
96.
— screw, 224.
— of rigid body, 225.
Moon, motion of, 20.
Motion, 3.
adiabatic, 189.
— change of, 22.
— concealed, 179.
— constrained, 41.
— differential equa-
tions of, 24.
— geometry, os, 3.
— harmonic, 35.
— — elliptic, 36.
irrotational, 520.
— isocyclic, 189.
— laws of, 520.
periodic, 35.
relative, 247.
relatively to earth,
320.
— steady, in hydrody-
namics ,
— uniform, 21.
— — circular, 37.
vortex, 509.
— of waves, 529.
Motus, 26.
Moving axes, 243.
— — motion with re-
spect to, 316.
Multipliers , indeterminate,
62.
Navier, 456.
Neumann, 388, 459, 515.
Newton, 3, 20, 29, 31, 66,
155, 365, 535, 544.
Newton's theorem, 4P9.
Newtonian constant, 30.
— law, force - function
for, 75.
— — motion under,39.
Node, 168, 169, 546.
Non- conservative system,
69.
Normal coordinates, 163.
— functions, 171.
Normal functions, series of,
173.
Null -system, 209.
Numeric, 26.
Nutation of top, 283.
Obry, 257.
Oersted's piezometer, 477.
Oscillation, axis of, 251.
— damped, 148.
Oscillations, small, 157.
Overtones, 169.
Pantheon, 324.
Parabola, path of projec-
tile, 34.
Parallelepiped, moment of
inertia of, 241.
Parameters of Lagrange,
111.
— differential, 88, 330,
344.
Parametric representation
of curve, 10.
Particle derivative, 497.
Path, 9.
Pendulum , affected by
earth's rotation, 323.
— compound, 251.
— conical, 55.
— cycloidal, 148.
— gyroscopic, 282.
— horizontal, 252.
ideal, 42.
— Eater's, 251.
— plane, 45.
— quadrantal, 196.
— small vibrations of,
54.
— spherical, 48.
— — by Lagrange's
equations.
path of, 50.
Perfect differential, 88.
— fluid, 458.
Period of pendulum, 45, 47.
Perpendicularity, condition
of, 7.
Phase, difference of, 37.
Picard, 86, 378, 388.
Piezometer, 477.
Pitch -conic, 220.
Pitch of helix, 211.
Pivoting, friction of, 304.
37*
586
INDEX.
Planet, motion of by Ha-
milton's method, 142.
* orbit of, 18.
period of, 19.
Plucker, 215.
Poincare, 105, 388, 393, 468.
Poinsot, 252, 255, 256, 261,
263.
— central axis, 210.
Point -function, 88, 329.
Point, material, 21.
Poiseuille, 552.
Poisson, 402, 491.
equation of, 359.
— for two dimen-
sions, 387.
ratio, 463.
Polar coordinates, 334.
Pole, motion of earth's, 270.
Polhode, 262.
cone, 263.
figure of, 263.
— projections of, 265.
Polygon of vectors, 5.
Polynomial of Legendre,
39.
Potential, definition of, 352.
characteristics of,
362.
of continuous distri-
bution, 353.
determined by pro-
perties, 375.
derivatives of, 355.
— development in
spherical harmonics,
402.
of disc, cylinder, and
cone, 366.
due to cylinder, 384.
— of earth's attraction,
298.
energy, apparent,
180.
of ellipsoid, 415.
— for internal
point, 418.
— — of revolution,
421.
— — — development
of, 424.
— differentiation
of, 419.
— of gravity, 405.
Potential, kinetic, 179.
— logarithmic, 385.
of sphere, 363.
of strain, 446.
— of tidal forces, 408.
vector, 511.
of vector, 89.
— velocity, 507.
Positional coordinates of
cyclic system 189.
Precession, 255.
— couple in regular,
274.
aijd nutation of top,
283.
— of earth, 298.
Pressure, 447.
Principal axes, of inertia,
229.
— coordinates, 163.
— function, 132.
Principle of Archimedes,
472.
of center of mass, 91.
of energy, 67.
of Hamilton, 97, 98.
— broader than
that of energy, 99.
— most general
principle, 130.
of least action, 97,
99, 101.
— of moment of mo-
mentum, 96.
- of varying action,! 31.
Problem of Dirichlet, 376.
— for circle, 388.
— for sphere, 395.
— of three bodies, 31.
Product, geometric, 7.
of inertia, 227.
scalar, 7.
— vector, 8.
Projections, 4.
of vector, 6.
Projectile, path of, 34.
Puiseux, 54, 299.
Pulley, 60.
Pump, Bunsen's, 504.
Pupin, 169, 170.
Pure strain, 430.
Quadrantal pendulum, 196.
Quadrics, confocal, 235.
Quadrics, elongation and
compression, 441.
— reciprocal, 431.
Ratio of Poisson, 463.
Rayleigh,123, 157, 169,529,
556.
.Reaction, 22.
accelerational , 122.
— of constraint, 42.
— — does no work, 65.
kinetic, 64, 119, 120.
— non- conservative,
122.
— static, 25.
Reciprocal distance, deve-
lopment of, 398.
— relations in cyclic
systems, 191.
quadrics, 431.
Reduction of groups of
forces, 209.
Reflection of wave, 545.
Relations between stress
and strain, 455.
Relative motion, 247.
Resal, 299.
Resistance of air, 26.
Resonance, 152.
— general theory of,175.
Resultant, 5
Richarz, 362.
Riemann, 378, 523.
Riemann -Weber, 393.
Righting couple, 473.
Rigid body, displacement
of, 200.
statics of, 205.
Rigidity, 462.
Rolling, 307.
— ellipsoid, 259.
— treated by Lagran-
ge's equations, 313.
Rotation, energy due to, 249.
— of earth, 298.
— momentum due to,
249.
of rigid body, 199.
— — — about in-
tersecting axes, 202.
— — about pa-
rallel axes, 201.
of rigid body, infini-
tesimal, 203.
INDEX.
587
Rotation, as vector, 203.
Roating axes,, 317.
— fluid, 467.
Routh, 176, 179.
Routh and Helmholtz, trans-
formation of, 177.
Sachse, 393.
de Saint Tenant, 478, 485.
— problem of, 478.
Scalars, 4.
Screw, momentum, 224.
Screws, composition of, 216.
— reciprocal, 221.
Schwerer, 296.
Searle, 269.
Second, mean solar, 27.
— differential parame-
ter, 344.
Self-conjugate function,430.
Sense, muscular, 24.
Series, hypergeometric, 312.
— trigonometric, 173,
388.
Sevres, 26.
Shear, 439.
amount of, 441.
general, 441.
— modulus of, 462.
Shifts, 436.
Ship, heeling of, 72.
Simple strains, 439.
stresses, 451.
Sleeping top, 304.
Slesser, 299.
Slides, 441.
Small oscillations, 157.
Solenoidal condition, 349.
vector, 347.
Solid angle, 351.
Somoff, 380.
Sound-waves, 542.
Sources and sinks, 520.
— strength of, 521.
Source of Sound, strength
of, 547.
Space, connectivity of, 339.
of m dimensions, 113.
Sphere, moment of inertia
of, 241.
— potential of, 365.
under pressure, 475.
Spherical harmonics, 393.
— — axes of, 396.
Spherical harmonics, deve-
lopment in, 400.
— — — of potential
in, 402.
— — forms of, 395.
— zonal, 397.
— waves, 547.
.Squeezes, 439.
Squirt, 521.
Statics, 57.
foundation of, 58.
— of rigid body, 205.
Steps, 4.
Steady motion in hydro-
dynamics, 503.
Stiffness, 151, 158.
Stokes, 84, 86, 345, 404,
497, 510, 533.
— theorem, 86.
— — in curvilinear
coordinates, 381.
Strain, axes of, 433.
ellipsoid, 432.
— general small, 436,
— heterogeneous, 444.
— homogeneous, 428.
— inverse, 432.
— irrotational, 433.
pure, 430.
— potential, 446.
relatively homoge-
neous, 445.
— composition of, 437.
Strength of field, 353.
Stress, 24, 446.
— ellipsoid, 451.
Cauchy's 451.
Lamp's, 451.
— geometrical represen-
tation of, 450.
— in isotropic bodies,
460.
— simple, 451.
— vector, 447.
work of, 454.
— and strain, relations
between, 455.
String, vibrations of, 164.
Surface distribution, 367.
— of equal action, 139.
level, 88.
— parallel, 139.
Sum, geometric, 5.
System, conservative, 65.
System, cyclic, 188.
— non-conservative, 69.
Tait, 141.
Target, problem of shoot-
ing, 103.
Tautochrone, 144.
Taylor's theorem, 36, 78,
79, 157, 346, 391,
399, 444, 455, 509.
Tensor of vector, 6.
Terrestrial gravity, poten-
tial of, 405.
Theorem, of bending mo-
ment, 490.
— Bernoulli's, 505.
— Chasles's, 413.
Clairaut's, 404.
of Coriolis, 319.
divergence, 347.
— Euler's, 114, 127.
— Gauss's, 350.
— for two dimen-
sions, 387.
— on energy, 425.
Green's, 340, 343.
— — for plane, 386.
— — in curvilinear
coordinates, 381.
— Hamilton's, 135.
— Helmholtz's, 612.
— Ivory's, 420.
for liquids, 465.
— Maclaurin's, 414.
of mean, Gauss's, 374.
— Newton's, 409.
of parallel axes, 229.
Taylor's, 36, 78, 79,
157, 346, 391, 399,
444, 455, 509.
Stokes's, 86.
— — in curvilinear
coordinates, 381.
— Thomson and Tait's,
138.
Torricelli's , 506.
Thomson and Tait, 21, 29,
72, 76, 105, 138, 148,
179, 196, 278, 378,
424, 459, 462, 470.
Tides, canal theory of, 539.
— equilibrium theory
of, 535.
— inversion of, 541.
588
INDEX.
Tide-generating forces, po-
tential of, 408.
Time, 3, 4.
measurement of, 22.
Tisserand, 470.
Todhunter and Pearson, 438,
459.
Tone, pure, 546.
Top, curve, 272.
heavy symmetrical,
274.
— Maxwell's, 268.
— kinetic reaction of,
271.
— rising of, 303.
rise and fall of, 285.
on smooth table, 302.
— nearly vertical, 289.
symmetrical, 271.
— equations by Jacobi's
method, 297.
Lagrange's me-
thod, 277.
Torricelli's theorem, 506.
Torsion, 483.
Traction, 447.
Translation of rigid body,
199.
- decomposed
into rotations, 202.
Trigonometric series, 388.
Trochoids drawn by top,288.
Torpedo, Whitehead, 257.
Howell, 272.
Triangle of vectors, 5.
Tube of vector-function, 348.
Tuning, 154.
Tuning-fork, 36.
Twist of billiard-ball, 306.
Uniform motion, 21.
circular motion, 37.
Units, absolute system of, 31 .
derived, 27.
— dimensions of, 27.
— C. G. S., 29.
Variable, complex, 521.
Variations, calculus of, 77.
differentiation of, 79.
— of integral, 80.
Varying action, 131.
— constraint, 129.
Vectors, 4.
addition of, 5.
couples of, 204.
free, 199.
lamellar, 87.
— product, 8.
— polygon of, 5.
sliding, 200.
tensor of, 6.
Vector -cross, 212.
— — conjugate lines
in, 213.
— function, linear, 428.
— potential, 511.
Velocityv-definition of, 9.
— angular, 11.
— components of, 10.
composition of, 10.
generalized, 112.
moment of, 12.
— sector, 12.
— transverse and radial,
11.
potential, 507.
— due to vortex, 514.
critical, for planet,40.
Venturi water-meter, 504.
Vibrations, energy of, 164.
forced, 152.
— free, 153.
— forced and free coex-
isting, 157.
isochronous necess-
arily harmonic, 146.
— normal, 163.
small, 36, 46.
Vierkant, 313, 314.
Virtual work, 57.
— — principle of, 61.
— displacement, 58.
Viscosity, 551.
Viscous fluids, 549.
— solid, 555.
Vortex, 506.
— couple, 520.
— energy of, 515.
— in corner, 519.
— ring, 620.
— straight, 515.
— strenght of, 507
— velocity due to,
Vortex -motion, 509.
— — conservati
511.
Vorticity, 502.
Water-meter, Ventu
Wave -motion, 529.
Waves in deep water,
— differential equat
of, 543.
— highest, 533.
— plane, 543.
— in shallow *•
535.
— in solid, 548.
— standing, 535
— of sound, 54.
Webster, 127, 381,
Weierstrass, 378.
Weight of kilogram
Whitehead torpedo
Wien, 499.
Woodward, 362.
Work, 56.
— of stress, 454.
— of wrench, 22
— unit of, 56.
— virtual, 57.
Wrench, 216.
— impulsive, 225.
— work of, 220.
Young's modulus, 462
Zollner, 252.
Zonal harmonics, 397.
ERRATA.
p. 30, line 21, for y = 6.576 read y = 6.6576.
„ 205, line 7 from bottom, for AB and PQ read AP and BQ.
Dresden.
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