Subject to this, the kinematical condition of the system, and

I denoting the length of the cord, and (u, v, w), (u', v', w), the components of the given velocities at its two ends: it is required to find x, y, ż at every point, so as to make

a minimum, μ denoting the mass of the string per unit of length, at the point P, which need not be uniform from point to point; and of course

Multiplying (a) by λ, an indeterminate multiplier, and proceeding as usual according to the method of variations, we have

dc đồi dy đồi đã đời
+ + )}ds=0,
ds ds ds ds ds ds

in which we may regard x, y, z as known functions of s, and this
it is convenient we should make independent variable. Inte-
grating "by parts" the portion of the first member which contains
A, and attending to the terminal conditions, we find, according to
the regular process, for the equations containing the solution

Generation of motion by impulse in an inextensible cord or

chain.

Supposing now, for simplicity, that s is independent variable, and performing the differentiation here indicated, with attention to

and the expression (§ 9) for p, the radius of curvature, we

Generation of motion

by impulse in an inextensible cord or chain.

The interpretation of (c) is very obvious. It shows that A is the impulsive tension at the point P of the string; and that the velocity which this point acquires instantaneously is the resultant

of

1 dλ

λ

ρμ

tangential, and towards the centre of curvature. μ ds The differential equation (e) therefore shows the law of transmission of the instantaneous tension along the string, and proves that it depends solely on the mass of the cord per unit of length in each part, and the curvature from point to point, but not at all on the plane of curvature, of the initial form. Thus, for instance, it will be the same along a helix as along a circle of the same

curvature.

With reference to the fulfilling of the six terminal equations, a difficulty occurs inasmuch as i, j, 2 are expressed by (d) immediately, without the introduction of fresh arbitrary constants, in terms of A, which, as the solution of a differential equation of the second degree, involves only two arbitrary constants. The explanation is, that at any point of the cord, at any instant, any velocity in any direction perpendicular to the tangent may be generated without at all altering the condition of the cord even at points infinitely near it. This, which seems clear enough without proof, may be demonstrated analytically by transforming the kinematical equation (a) thus. Let f be the component tangential velocity, q the component velocity towards the centre of curvature, and p the component velocity perpendicular to the osculating plane. Using the elementary formulæ for the direction cosines of these lines (§ 9), and remembering that s is now independent variable, we have

i=fdx+qpd2x+pp(dzd3y—dyd3z)

ds ds2

, y = etc.

dss

a form of the kinematical equation of a flexible line which will be of much use to us later.

We see, therefore, that if the tangential components of the impressed terminal velocities have any prescribed values, we may give besides, to the ends, any velocities whatever perpendicular to the tangents, without altering the motion acquired by any part of the cord. From this it is clear also, that the directions of the terminal impulses are necessarily tangential; or, in other words,

that an impulse inclined to the tangent at either end, would Generation generate an infinite transverse velocity.

of motion by impulse in an in

cord or

To express, then, the terminal conditions, let F and F" be the extensible tangential velocities produced at the ends, which we suppose chain. known. We have, for any point, P, as we have seen above,

which suffice to determine the constants of integration of (d). Or if the data are the tangential impulses, I, I', required at the ends to produce the motion, we have

Or if either end be free, we have λ=0 at it, and any prescribed condition as to impulse applied, or velocity generated, at the other end.

The solution of this problem is very interesting, as showing how rapidly the propagation of the impulse falls off with "change of direction" along the cord. The reader will have no difficulty in illustrating this by working it out in detail for the case of a

the form of a circle or helix. The results have curious, and dynamically most interesting, bearings on the motions of a whip lash, and of the rope in harpooning a whale.

motion of

sible liquid

Example (3). Let a mass of incompressible liquid be given at Impulsive rest completely filling a closed vessel of any shape; and let, by incompres suddenly commencing to change the shape of this vessel, any arbitrarily prescribed normal velocities be suddenly produced in the liquid at all points of its bounding surface, subject to the condition of not altering the volume: It is required to find the instantaneous velocity of any interior point of the fluid.

Let x, y, z be the co-ordinates of any point P of the space occupied by the fluid, and let u, v, w be the components of the generated velocity of the fluid at this point. Then p being the density of the fluid, and denoting integration throughout the space occupied by the fluid, we have

Impulsive motion of incompressible liquid.

which, subject to the kinematical condition (§ 193),

du dv dw
+ +
dx dy dz

must be the least possible, with the given surface values of the
normal component velocity. By the method of variation we have
dou dov, dow
♫♫S{p(udu+v&v+w&w) +λ(dx + + dxdydz=0. (c)
dy dz

But integrating by parts we have

đâu . đôi

đầu

+ + )dxdydz=ffλ(Sudydz+8vdzdx+Swdxdy)

dx dy

dz

and if l, m, n denote the direction cosines of the normal at any point of the surface, dS an element of the surface, and tegration over the whole surface, we have

in

Sudydz+8vdzdx+8wdxdy)=ƒƒλ(lôu+mdv+ndw)dS=0,

since the normal component of the velocity is given, which requires that lou+mdv+n&w=0. Using this in (c), (d), and equating the coefficients of Su, Sv, dw, we have

an equation for the determination of A, whence by (e) the

solution is completed.

The condition to be fulfilled, besides the kinematical equation (b), amounts to this merely, that p(udx+vdy+wdz) must be a complete differential. If the fluid is homogeneous, p is constant, and udx+vdy+wdz must be a complete differential; in other words, the motion suddenly generated must be of the "non-rotational" character [§ 190, (i)] throughout the fluid mass. The equation to determine à becomes, in this case,

From the hydrodynamical principles explained later it will appear that A, the function of which pudx+vdy+wd:) is the differential, is the impulsive pressure at the point (x, y, z) of the fluid. Hence we may infer that the equation (ƒ), with the condition that A shall have a given value at every point of a certain closed surface, has a possible and a determinate solution for every point within that surface. This is precisely

motion of

the same problem as the determination of the permanent tempe- Impulsive
rature at any point within a heterogeneous solid of which the incompres
surface is kept permanently with any non-uniform distribution sible liquid.
of temperature over it, (f) being Fourier's equation for the
uniform conduction of heat through a solid of which the conduct-

Ρ

determinateness of this problem were both proved above [Chap. 1.
App. A, (e)] by a demonstration, the comparison of which
with the present is instructive. The other case of superficial
condition that with which we have commenced here-shows
αλ dλ αλ

that the equation (f), with l +m + n given arbitrarily
dx dy dz

for every point of the surface, has also a possible and single
solution for the whole interior space. This, as we shall see in
examining the mathematical theory of magnetic induction, may
also be inferred from the general theorem (e) of App. A above,
by supposing a to be zero for all points without the given surface,

and to have the value for any internal point (x, y, z).

1 P

318. Maupertuis' celebrated principle of Least Action has Least action. been, even up to the present time, regarded rather as a curious and somewhat perplexing property of motion, than as a useful guide in kinetic investigations. We are strongly impressed with the conviction that a much more profound importance will be attached to it, not only in abstract dynamics, but in the theory of the several branches of physical science now beginning to receive dynamic explanations. As an extension of it, Sir W. R. Hamilton' has evolved his method of Varying Action, which undoubtedly must become a most valuable aid in future generalizations.

What is meant by "Action" in these expressions is, unfor- Action. tunately, something very different from the Actio Agentis defined by Newton, and, it must be admitted, is a much less judiciously chosen word. Taking it, however, as we find it, now universally used by writers on dynamics, we define the Action of a Moving System as proportional to the average Time aver kinetic energy, which it has possessed for the time from any energy. convenient epoch of reckoning, multiplied by the time. Ac

1 Phil. Trans. 1834-1835.

age of