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

when

and when

8 = 0, λ = I,)

8 = 1, λ = I'S

.(i).

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

is constant, and given in

When μ and p are constant,

the form of a circle or helix.
for instance, the impulsive tension decreases in the proportion
of 1 to per space along the curve equal to p. The results have
curious, and dynamically most interesting, bearings on the mo-
tions of a whip lash, and of the rope in harpooning a whale.

Generation

of motion by impulse in an inextensible cord or

chain.

motion of

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 sible liquid. 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 required velocity of the fluid at this point. Then p being the density of the fluid, and ƒfƒ denoting integration throughout the space occupied by the fluid, we have

T= ƒƒƒ }p (u2 + v3 + w3) dxdydz .....

(a),

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

must be the least possible, with the given surface values of the normal component velocity. By the method of variation we have

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 ƒƒ integration over the whole surface, we have

♫♫λ (Sudydz + dvdzdx + Swdxdy) = ƒƒλ (lòu + mdv + ndw) dS = 0, since the normal component of the velocity is given, which requires that lầu + m&v + n&w=0. Using this in going back with the result to (c), (d), and equating to zero the coefficients of δκ, δν, δι, we find

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 A becomes, in this case,

motion of

From the hydrodynamical principles explained later it will Impulsive appear that λ, the function of which p(udx + vdy+wdz) is incompres sible líquid. 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 the same problem as the determination of the permanent temperature at any point within a heterogeneous solid of which the surface is kept permanently with any non-uniform distribution of temperature over it, (f) being Fourier's equation for the uniform conduction of heat through a solid of which the conducting power at the point (x, y, z) is

1

The possibility and the

determinateness of this problem (with an exception regarding
multiply continuous spaces, to be fully considered in Vol. II.)
were both proved above [Chap. 1. App. A, (e)] by a demonstra-
tion, the comparison of which with the present is instructive.
The other case of superficial condition-that with which we
have commenced here-shows that the equation (f), with
dλ αλ

dx

+ m

+ n

dy dz

given arbitrarily for every point of the sur

face, has also (with like qualification respecting multiply con-
tinuous spaces) a possible and single solution for the whole
interior space.
This, as we shall see in examining the mathe-
matical 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

equations of

terms of

co-ordinates

318. The equations of continued motion of a set of free Lagrange's particles acted on by any forces, or of a system connected in motion in any manner and acted on by any forces, are readily obtained generalized in terms of Lagrange's Generalized Co-ordinates by the regular and direct process of analytical transformation, from the ordinary forms of the equations of motion in terms of Cartesian (or rectilineal rectangular) co-ordinates. It is convenient first to effect the transformation for a set of free particles acted on by any forces. The case of any system with invariable connexions, or with connexions varied in a given manner, is

deduced
direct by
transforma-
tion from
the equa
tions of
motion in
terms of
Cartesian
co-ordi-

nates.

then to be dealt with by supposing one or more of the generalized co-ordinates to be constant: or to be given functions of the time. Thus the generalized equations of motion are merely those for the reduced number of the co-ordinates remaining un-given; and their integration determines these co-ordinates.

Let m1, m2, etc. be the masses, r1, y1, z1, x,, etc. be the coordinates of the particles; and X,, Y1, Z,, X,, etc. the components of the forces acting upon them. Let 4, 4, etc. be other variables equal in number to the Cartesian co-ordinates, and let there be the same number of relations given between the two sets of variables; so that we may either regard 4, 4, etc. as known functions of x, y, etc., or x, y,, etc. as known functions of ,, etc. Proceeding on the latter supposition we have the equations (a), (1), of § 313; and we have equations (b), (6), of the same section for the generalized components V, 4, etc. of the force on the system.

For the Cartesian equations of motion we have

dx. X,=m, dťa

2

etc....(19).

Using this and similar expressions with reference to the other co-ordinates in (20), and remarking that

2

{m, (x2 + ÿ,2 + ¿,2) + 1 m, (etc.) + etc. T.........(22),

if, as before, we put 7 for the kinetic energy of the system; we

d

di

die

=

dt

dy

d1d (x,2)
dy

(21).

dy

The substitutions of

di, for dy

Lagrange's equations of motion in terms of generalized co-ordinates

direct by

tion from

above, suppose to be a function of the co-ordinates, and of the generalized velocity-components, as shown in equations (1) of deduced § 313. It is on this supposition [which makes T a quadratic transformafunction of the generalized velocity-components with functions the equaof the co-ordinates as coefficients as shown in § 313 (2)] that the motion in d d differentiations and in (23) are performed. Proceeding Cartesian di dy

similarly with reference to 4, etc., we find expressions similar to (23) for 4, etc., and thus we have for the equations of motion in terms of the generalized co-ordinates

tions of

terms of

It is to be remarked that there is nothing in the preceding transformation which would be altered by supposing t to appear in the relations between the Cartesian and the generalized coordinates: thus if we suppose these relations to be

co-ordinates.

dx

where

dt

denotes what the velocity-component, would be

if y, ø, etc. were constant; being analytically the partial differential coefficient with reference to t of the formula derived from (26) to express x, as a function of t, 4, 4, 0, etc.

Using (26) in (22) we now find instead of a homogeneous quadratic function of 4, 4, etc., as in (2) of § 313, a mixed