Equation of
Energy.

The equation of energy, found as above [§ 318 (29iv) and (29)], is

The interpretation, considering (12), is obvious. The contrast with Example F' (g) is most instructive.

Sub-Example (G1).—Take, from above, Example C, case (a): and put =+0; also, for brevity, mj3+na2=B, n(b2+k2) = A, and nab = c. We have*

T = § {A↓3 + 2c↓ († + 0) cos 0 + B (↓ + Ô)3} ;

Here the co-ordinate 0 alone, and not the co-ordinate y, appears in the coefficients. Suppose now = 0 [which is the case con

sidered at the end of C (a) above]. We have

dT

dy

=

C, and

* Remark that, according to the alteration from 4, 4, 4, 4, to 4, 4, 8, è, as independent variables,

which is to be fully integrated first by multiplying by de and integrating once; and then solving for dt and integrating again with respect to 0. The first integral, being simply the equation of energy integrated, is [Example G (20)]

Equation of
Energy.

of co

In the particular case in which the motion commences from Ignoration rest, or is such that it can be brought to rest by proper applica- ordinates. tions of force-components, V, P, etc. without any of the forcecomponents X, X', etc., we have C=0, C'= 0, etc.; and the elimination of x, x', etc. by (3) renders T a homogeneous quadratic function of 4, 4, etc. without C, C', etc.; and the equations of motion become

We conclude that on the suppositions made, the elimination of the velocity-components corresponding to the non-appearing coordinates gives an expression for the kinetic energy in terms of the remaining velocity-components and corresponding coordinates which may be used in the generalised equations just as if these were the sole co-ordinates. The reduced number of equations of motion thus found suffices for the determination of the co-ordinates which they involve without the necessity for knowing or finding the other co-ordinates. If the farther question be put,-to determine the ignored co-ordinates, it is to be answered by a simple integration of equations (7) with C = 0, C' = 0, etc.

One obvious case of application for this example is a system in which any number of fly wheels, that is to say, bodies which are

Ignoration

of co-ordinates.

kinetically symmetrical round an axis (§ 285), are pivoted frictionlessly on any moveable part of the system. In this case with the particular supposition C=0, C=0, etc., the result is simply that the motion is the same as if each fly wheel were deprived of moment of inertia round its bearing axis, that is to say reduced to a line of matter fixed in the position of this axis and having unchanged moment of inertia round any axis perpendicular to it. But if C, C', etc. be not each zero we have a case embracing a very interesting class of dynamical problems in which the motion of a system having what we may call gyrostatic links or connexions is the subject. Example (D) above is an example, in which there is just one fly wheel and one moveable body on which it is pivoted. The ignored co-ordinate is ; and supposing now to be zero, we have

If we suppose C = 0 all the terms having ' for a factor vanish and the motion is the same as if the fly wheel were deprived of inertia round its bearing axis, and we had simply the motion of the "ideal rigid body PQ" to consider. But when C does not vanish we eliminate from the equations by means of (a). It is important to remark that in every case of Example (G) in which C=0, C'= 0, etc. the motion at each instant possesses the property (§ 312 above) of having less kinetic energy than any other motion for which the velocity-components of the non-ignored co-ordinates have the same values.

Take for another example the final form of Example C' above, putting B for C, and A for nk2 + A. We have

T = {{(E + F cos2 0) ↓2 + B (↓ cos 0 + $)2 + AÒ3} ...(22). Here neither nor appears in the coefficients. Let us suppose • = 0, and eliminate ¿, to let us ignore 4. We have

The place of x in (9) above is now taken by 4, and comparing

Hence, and as K is constant, the equations of motion (19) Ignoration

become

of co

ordinates.

A most important case for the "ignoration of co-ordinates" is presented by a large class of problems regarding the motion of frictionless incompressible fluid in which we can ignore the infinite number of co-ordinates of individual portions of the fluid and take into account only the co-ordinates which suffice to specify the whole boundary of the fluid, including the bounding surfaces of any rigid or flexible solids immersed in the fluid. The analytical working out of Example (G) shows in fact that when the motion is such as could be produced from rest by merely moving the boundary of the fluid without applying force to its individual particles otherwise than by the transmitted fluid pressure we have exactly the case of C=0, C'0, etc.: and Lagrange's generalized equations with the kinetic energy expressed in terms of velocity-components completely specifying the motion of the boundary are available. Thus,

a perfect

320. Problems in fluid motion of remarkable interest and Kinetics of importance, not hitherto attacked, are very readily solved by liquid. the aid of Lagrange's generalized equations of motion. For brevity we shall designate a mass which is absolutely incompressible, and absolutely devoid of resistance to change of shape, by the simple appellation of a liquid. We need scarcely say that matter perfectly satisfying this definition does not exist in nature: but we shall see (under properties of matter) how nearly it is approached by water and other common real liquids. And we shall find that much practical and interesting information regarding their true motions is obtained by deductions from the principles of abstract dynamics applied to the ideal perfect liquid of our definition. It follows from Example

a perfect

liquid.

Kinetics of (G) above (and several other proofs, some of them more synthetical in character, will be given in our Second Volume,) that the motion of a homogeneous liquid, whether of infinite extent, or contained in a finite closed vessel of any form, with any rigid or flexible bodies moving through it, if it has ever been at rest, is the same at each instant as that determinate motion (fulfilling, § 312, the condition of having the least possible kinetic energy) which would be impulsively produced from rest by giving instantaneously to every part of the bounding surface, and of the surface of each of the solids within it, its actual velocity at that instant. So that, for example, however long it may have been moving, if all these surfaces were suddenly or gradually brought to rest, the whole fluid mass would come to rest at the same time. Hence, if none of the surfaces is flexible, but we have one or more rigid bodies moving in any way through the liquid, under the influence of any forces, the kinetic energy of the whole motion at any instant will depend solely on the finite number of coordinates and component velocities, specifying the position and motion of those bodies, whatever may be the positions reached by particles of the fluid (expressible only by an infinite number of co-ordinates). And an expression for the whole kinetic energy in terms of such elements, finite in number, is precisely what is wanted, as we have seen, as the foundation of Lagrange's equations in any particular case.

It will clearly, in the hydrodynamical, as in all other cases, be a homogeneous quadratic function of the components of velocity, if referred to an invariable co-ordinate system; and the coefficients of the several terms will in general be functions of the co-ordinates, the determination of which follows immediately from the solution of the minimum problem of Example (3) § 317, in each particular case.

Example (1).-A ball set in motion through a mass of incompressible fluid extending infinitely in all directions on one side of an infinite plane, and originally at rest. Let x, y, z be the coordinates of the centre of the ball at time t, with reference to rectangular axes through a fixed point O of the bounding plane, with OX perpendicular to this plane. If at any instant either