where y,, ... denote generalized components of the continuous force acting at any instant of the infinitely short time 7, within which the impulse is completed. generation referred to If this impulse is applied to the system, previously in motion Impulsive in the manner specified above, and if d4, 84, ... denote the re- of motion sulting augmentations of the components of velocity, the means generalized of the component velocities before and after the impulse will be nates. † + 184, $+18$, ...... Hence, according to the general principle explained above for calculating the work done by an impulse, the whole work done. in this case is ¥ († + 1 84) + ☀ ($ + 184) + etc. To avoid unnecessary complications, let us suppose di, 84, etc., to be each infinitely small. The preceding expression for the work done becomes ¥ + + etc.; and, as the effect produced by this work is augmentation of kinetic energy from T to T + ST, we must have ST= ↓ + + etc. Now let the impulses be such as to augment & to + dỷ, and to leave the other component velocities unchanged. We shall have co-ordi to the coefficients of 4, 4, ... respectively in dy (c) From this we see, further, that the impulse required to produce the component velocity from rest, or to generate it in the system moving with any other possible velocity, has for its components (4, 4) 4, (4, 4) 4, (4, 0) į, etc. Hence we conclude that to generate the whole resultant velocity VOL. I. Momenta in terms of velocities. Kinetic energy in terms of momentums and velocities. where it must be remembered that, as seen in the original expression for T, from which they are derived, (4, 4) means the same thing as (4, 4), and so on. The preceding expressions are the differential coefficients of T with reference to the velocities; that is to say, (d) The second members of these equations being linear functions of 4, 4, ..., we may, by ordinary elimination, find 4, 4, etc., in terms of έ, 7, etc., and the expressions so obtained are of course linear functions of the last-named elements. And, since T is a quadratic function of 4, 4, etc., we have Now the algebraic process by which 4, 4, etc., are obtained in in terms of These expressions solve the direct problem,-to find the velo- Velocities city produced by a given impulse (έ, 7, ...), when we have the momenkinetic energy, T, expressed as a quadratic function of the components of the impulse. (e) If we consider the motion simply, without reference to the impulse required to generate it from rest, or to stop it, the quantities έ, ŋ, ... are clearly to be regarded as the components of the momentum of the motion, according to the system of generalized co-ordinates. (f) The following algebraic relation will be useful: .... tums. Reciprocal relation momentums ties in two §‚† + n‚$ + ¿‚0 + etc. = §į, + n$, + 30, + etc................................ (11), between where, έ, n, 4, 4, etc., having the same signification as before, and veloci §, n,, §,, etc., denote the impulse-components corresponding to motions. any other values,,,,,,, etc., of the velocity-components. It is proved by observing that each member of the equation becomes a symmetrical function of 4, 4; 4, 4,; etc. ; when for έ,, 7,, etc., their values in terms of 4,, ¿,, etc., and for έ, ŋ, etc., their values in terms of 4, 4, etc., are substituted. Application ized co 314. A material system of any kind, given at rest, and subjected to an impulse in any specified direction, and of any of generalgiven magnitude, moves off so as to take the greatest amount ordinates of kinetic energy which the specified impulse can give it, of 811. subject to § 308 or § 309 (c). Let έ, n, į, ¿, ... ... be the components of the given impulse, and the components of the actual motion produced by it, which are determined by the equations (10) above. Now let us suppose the system be guided, by means of merely directive constraint, to take, from rest, under the influence of the given. impulse, some motion (,,,,...) different from the actual motion; and let έ,, 7,, ... be the impulse which, with this constraint removed, would produce the motion (4,, &,, ...). We shall have, for this case, as above, But -,,-n T‚ = § (§‚†, + n‚Å‚ + ...). ... are the components of the impulse experienced in virtue of the constraint we have supposed introduced. They neither perform nor consume work on the system when moving as directed by this constraint; that is to say, (§, − §) ↓, + (n, − n) 4, + (5, − 5) 6, + etc. = 0...........(12); to theorems Theorems of § 311 in terms of generalized co-ordi nates. Problems whose data involve impulses and velocities. §, (į − ¿,) + n, (¿ − ¿,) + etc. = (§ — §,) 4, + (n − n,) 4, + etc. = 0, and therefore we have finally 2 ( T − T) = (§ — §,) (↓ − ¿,) + (n − n,) ($ − ¿,) + etc. ... (14), that is to say, T exceeds T, by the amount of the kinetic energy that would be generated by an impulse (§ — §,, n − 1, §−3,, etc.) applied simply to the system, which is essentially positive. In other words, 315. If the system is guided to take, under the action of a given impulse, any motion (,,,, ...) different from the natural motion (4, $, ...), it will have less kinetic energy than that of the natural motion, by a difference equal to the kinetic energy of the motion ( ýˆ‚‚ $ — $,, ...). COR. If a set of material points are struck independently by impulses each given in amount, more kinetic energy is generated if the points are perfectly free to move each independently of all the others, than if they are connected in any way. And the deficiency of energy in the latter case is equal to the amount of the kinetic energy of the motion which geometrically compounded with the motion of either case would give that of the other. (a) Hitherto we have either supposed the motion to be fully given, and the impulses required to produce them, to be to be found; or the impulses to be given and the motions produced by them to be to be found. A not less important class of problems is presented by supposing as many linear equations of condition between the impulses and components of motion to be given as there are degrees of freedom of the system to move (or independent co-ordinates). These equations, and as many more supplied by (8) or their equivalents (10), suffice for the complete solution of the problem, to determine the impulses and the motion. whose data pulses and (b) A very important case of this class is presented by prescrib- Problems ing, among the velocities alone, a number of linear equations with involve imconstant terms, and supposing the impulses to be so directed and velocities. related as to do no work on any velocities satisfying another prescribed set of linear equations with no constant terms; the whole number of equations of course being equal to the number of independent co-ordinates of the system. The equations for solving this problem need not be written down, as they are obvious; but the following reduction is useful, as affording the easiest proof of the minimum property stated below. (c) The given equations among the velocities may be reduced. to a set, each homogeneous, except one equation with a constant term. Those homogeneous equations diminish the number of degrees of freedom; and we may transform the co-ordinates so as to have the number of independent co-ordinates diminished accordingly. Farther, we may choose the new co-ordinates, so that the linear function of the velocities in the single equation with a constant term may be one of the new velocity-components; and the linear functions of the velocities appearing in the equation connected with the prescribed conditions as to the impulses may be the remaining velocity-components. Thus the impulse will fulfil the condition of doing no work on any other component velocity than the one which is given, and the general problem— problem § 312). 316. Given any material system at rest: let any parts of General it be set in motion suddenly with any specified velocities, pos- (compare sible according to the conditions of the system; and let its other parts be influenced only by its connexions with those; required the motion: takes the following very simple form :-An impulse of the character specified as a particular component, according to the generalized method of co-ordinates, acts on a material system; its amount being such as to produce a given velocity-component of the corresponding type. It is required to find the motion. The solution of course is to be found from the equations ↓ = A, n = 0, ¿ = 0............. ..(15) (which are the special equations of condition of the problem) and the general kinetic equations (7), or (10). Choosing the latter, and denoting by [, ], [, n], etc., the coefficients of, n, etc., |