SUPPL

SUPPL. VOL. I. PART II.

some point C of the line PG, and will also be perpendicular to the same plane. All this has been demonstrated in the article ROTATION, n° 94. &c. Complete the parallelogram AFHE. It is plain, that the motion AH is equivalent to AE and AF. By the motion AE, A only slides along the surface of B without pressing it, or causing any tendency to motion in that direction, except perhaps a little arising from friction. It is by the motion AF alone that the impulse is made.

Therefore let $v$ be $V\; \backslash times\; \backslash frac\{AF\}\{AH\}$; and then $A\; \backslash times\; v$ may be called the efficient impulse of the body A in the present circumstances, and $v$ the efficient velocity. This will be diminished by the collision. Let $x$ be the unknown velocity remaining in A after the collision, or rather in the instant of the greatest compression and common motion of the touching points of A and B, estimated in the direction FP. The effective momentum lost by A must therefore be $A\; \backslash times\; v\; -\; x$; but the same must be gained by B, and its centre G must move in the direction GI, parallel to FP, with this momentum; and therefore with the velocity $\backslash frac\{A\; \backslash times\; v\; -\; x\}\{B\}$. That this

may be the case, the point of percussion F must yield with the velocity $x$, because the bodies are in contact. But because C is the spontaneous axis of conversion, every particle is beginning to describe an arch of a circle round this axis. Therefore F is beginning to move in the direction FG, perpendicular to the momentary radius vector CF. Let FG be a very minute arch, described in a moment of time. Draw $gf$ perpendicular to FP. Then $fg$ is the motion FG reduced to the direction FP, and will express the yielding of B in the direction of the impulse, while G describes a space equal to $\backslash frac\{A\; \backslash times\; v\; -\; x\}\{B\}$, and A describes a space $x$. There-

fore FG will express $x$. Let $Pp$ be the space described in the same time that FG is described. Draw $pC$, cutting GK in the point I. GI is the yielding of the body B to the impulse, and must therefore be equal to $\backslash frac\{A\; \backslash times\; v\; -\; x\}\{B\}$.

The triangles $Ffg$ and $CPF$ are similar; for the angle CFP is the complement of $fFg$ to a right angle: It is also the complement of PCF to a right angle. Therefore $Fg\; :\; Ff\; =\; FC\; :\; CP$. But $Fg\; :\; Pp\; =\; FC\; :\; CP$; because the little arches $Fg$, $Pp$ have the same angle at C. Therefore $Pp\; =\; Ff\; =\; x$. It is plain, that $CG\; :\; CP\; =\; GI\; :\; Pp$. Therefore $CG\; :\; CP\; =\; \backslash frac\{A\; \backslash times\; v\; -\; x\}\{B\}\; :\; x$, and $x\; =\; \backslash frac\{A\; \backslash times\; v\; -\; x\; \backslash times\; CP\}\{B\; \backslash times\; CG\}$, or $x\; =\; v\; \backslash times\; \backslash frac\{A\; \backslash times\; CP\}\{B\; \backslash times\; CG\}\; -\; x\; \backslash times\; \backslash frac\{A\; \backslash times\; CP\}\{B\; \backslash times\; CG\}$; wherefore

$x\; \backslash times\; B\; \backslash times\; CG\; +\; x\; \backslash times\; A\; \backslash times\; CP\; =\; v\; \backslash times\; A\; \backslash times\; CP$, and $x\; \backslash times\; B\; \backslash times\; CG\; +\; A\; \backslash times\; CP\; =\; v\; \backslash times\; A\; \backslash times\; CP$, and $x\; =\; v\; \backslash times\; \backslash frac\{A\; \backslash times\; CP\}\{B\; \backslash times\; CG\; +\; A\; \backslash times\; CP\}\; =$ the velocity remaining in A, estimated in the direction FP.

And $u$, the velocity with which G will advance, is $\backslash frac\{CG\}\{CP\}$; for $CP\; :\; CG\; =\; Pp\; :\; GI\; =\; x\; :\; u$. It is evident that A will change its direction by the collision: For in the instant of greatest compression, it was react-

Impulsion. ed on by a force $=\; A\; \backslash times\; v\; =\; x$ in the direction FA. This must be compounded with $A\; \backslash times\; V$, in the direction AH, in order to obtain the new motion of A; or it may be found by compounding $x$, which is retained by A, with FH, which has suffered no change by the collision. The bodies will therefore separate, although they be unelastic: If they are elastic, we must double these changes on each. If B was also in motion before the collision, the motion of A must be resolved into two, one of which is equal and parallel to the motion of B: the other must be employed as we have employed the motion AH.

Expressions still more general may be obtained for $x$ and $u$; namely, by taking the formulæ for the centres of conversion and percussion (ROTATION, n° 96, 99.)

where $p$ stands for a particle of matter, and $r$ for its distance from an axis passing through G perpendicularly to the plane of the lines GP and PF. In this way

It is plain from this proposition, that the progressive motion of the body depends, not only on the momentum of the impelling body, but also on the place where the other is struck: For even although the original momentum of A be the same, and the obliquity of the stroke, making $v$ the same, and the body (and consequently $\backslash int\; p\; r^2$) also remain the same, we see that $x$ and $u$ depend on the ratio of CP to CG. Now C and P are always on opposite sides of G: Consequently, by removing the direction FP of the impulsion farther from G, we diminish CG and increase CP; and therefore

increase the value of $x\; =\; v\; \backslash frac\{A\; \backslash cdot\; CP\}\{B\; \backslash cdot\; CG\; +\; A\; \backslash cdot\; CP\}$; and

consequently diminish the value of $A\; \backslash times\; v\; =\; x$, to which $B\; \backslash times\; u$ is equal. The greatest momentum of B is produced when the direction of the impulse passes through G, and no rotation is produced. Indeed we are led, by a sort of common sense, to expect this.

This investigation is by no means a piece of mere speculative curiosity. It is the solution of the greatest problem in practical mechanics. It is in this way that we must proceed in computing the actions of the wind and water on the sails and hull of a ship. Were it not that many circumstances concur in determining several of the preparatory steps, it is evident that the task must be almost impracticable. But the pressure and its direction are generally determined by experiment, without the trouble of computation; and we are seldom solicitous about the subsequent motion of the wind or water.

There is another question in impulsion which is of the first practical importance—namely, when the impulse is exerted on the parts of a machine, where the body struck is not at liberty to yield freely to the stroke, but must slide along some solid path, or turn round some axis, or take some other constrained motion. The operations of most engines depend on this. The operation of wedges, axes, and many cutting and piercing instruments, and the penetration of piles, im-

pelled by a rammer, are all ascertained by the same Impulsion doctrines. But the particular applications can scarcely be elucidated by any classification that occurs to us, the circumstances of the case making such great difference in the result, both in kind and degree. For example, in the simplest case that occurs, the driving of piles, the penetration of the pile depends, in the first place, on the momentum of the rammer. If the mass of the pile be neglected, the penetration through a uniformly resisting substance will be as the square of the velocity of the rammer, (DYNAMICS, Suppl. n° 95), and its absolute quantity may be determined from a knowledge of the proportion of the weight of the rammer to the resistance of the earth. But when we consider that we have to put in motion the whole matter of the pile, we learn that a great diminution of the effect must take place. We still can compute what this must be, because we have the same momentum, with a velocity diminished in a certain proportion of the sum of the matter in the rammer and pile, to that in the rammer alone.—Another defalcation arises from friction, which continually increases as the pile goes deeper;—and a still greater defalcation proceeds from the nature of the pile. If it is a piece of very dry straight grained fir, it is very elastic, and acquires almost a double velocity from the stroke of a rammer of cast iron. If it is moist and soft, especially if it is oak, or other timber of an undulated fibre, it does not acquire so great velocity, and the penetration is very much diminished. It is probable that a pile, headed with moist cork, could not be driven at all. The writer of this article found a remarkable effect of the elasticity in the process of boring limestone. When the boring bit was made entirely of steel, and tempered through its whole length to a hard spring temper, the workman bored three inches, in the same time that another bored two inches with a bit made of soft iron; and he would never use any but steel bits, if they could be hindered from chipping by the hammer (which must also be of tempered steel throughout). This has hitherto baffled many attempts. A pretty large round head, like a marlin spike, has succeeded: but even this cracks after some days use. The improvement is richly worth attention; for the workman is delighted by feeling the hammer rise in his hand after every stroke, and says that the work is not so hard by half. N.B. The stone cutters at Lisbon and Oporto use iron mallets.

76
The case of impulsion made on part of a machine moveable round an axis has been considered in the article ROTATION, Encycl. n° 72; where $x$ is shewn to be $=\; v\; \backslash times\; \backslash frac\{A\; \backslash cdot\; CP^2\}\{\backslash int\; p\; r^2\; +\; A\; \backslash cdot\; CP^2\}$. But, in this formula, $r$ denotes the distance of $p$ from the point C, and not from G. $\backslash int\; p\; r^2$ in this formula, is $B\; \backslash cdot\; CG\; \backslash cdot\; CP$; whereas, in the formula for a free body, where $r$ is the distance of a particle from G, $\backslash int\; p\; r^2$ is $=\; B\; \backslash cdot\; CG\; \backslash cdot\; GP$.

In the practical consideration of this question, the reader will do well to consider the whole of that article with attention. Many circumstances occur, which make a proper choice of the point of impulse, and the direction of the tangent plane, of the greatest consequence to the good performance of the machine; and there

Impulsion. there is nothing in which the scientific knowledge of the engineer is of more essential service to him. An engineer of great practice, and a sagacious combining mind, collects his general observations, and stores them up as rules of future practice. But it is seldom that he possesses them with that distinctness and confidence that can enable him to communicate his knowledge to others, or even secure himself against all mistakes; whereas a moderate acquaintance with these elements of real mechanics, may be applied with safety on all occasions, because arithmetical computations, when rightly made, afford the most certain of all results.

⁷⁷ Great loss of power by the yielding and bending of the materials.

There is a circumstance which greatly affects the performance of machines which are actuated by impulses, namely, the yielding and bending of the parts. When the moving power acts by repeated small impulses, it may sometimes be entirely consumed, without producing any effect whatever at the remote working point of the machine; and the engineer, who founds his constructions on the elementary theories to be had in most treatises of mechanics, will often be miserably disappointed. In the usual theories, even as delivered by writers of eminence, it is asserted, that the smallest impulse will start the greatest weight. But since impulse is only a continued pressure, and requires time for the transmission of its effect through the parts of a yielding solid, it is plain that the motion of the impelling body may be extinguished before it has produced compression enough for exciting the forces which are to raise the remote parts of a heavy body from the ground. What blow with a hammer could start a feather bed? Much oftener may we expect, that a blow, given to one arm of a long lever, will be consumed in bending the whole of its length, so as to bring the remote end into action. Therefore great stiffness, and perfect elasticity, both in the moving parts and in the points of support, are necessary for transmitting the full, or even a considerable part, of the power of the impelling body. Perhaps not the half of the blow given by the wipers of a great forge or tilting mill to the shank of the hammer is transmitted in the proper instant of time to the hammer-head. The hammer, while it is tossed up by the blow, is quivering as it flies. Should it reach the spring above it in the time of its downward vibration, it will not be returned with such force as if it had hit the spring a moment before or after. A quarter of an inch will produce a great effect in such cases. It is found, that the minute impulses given to the pallets of a clock or watch lose much of their force by the imperfect elasticity of the pendulum or balance. We must therefore make all the parts which transmit the blow to the regulating mass of matter as continuous, hard, elastic, and stiff, as possible. The performance of ruby pallets is very sensibly weakened by putting oil on the face of them, especially in the detached escapements, which act partly by impulse. A wheel of hard tempered steel, working on a dry ruby pallet, excels all others. The intelligent engineer, seeing that, after all his care, much impulse is unavoidably lost, will avoid employing a first mover which acts in a subalternary manner, and will substitute one of continued pressure when it is in his power. This is one chief cause of the great superiority of overshot water-wheels above the undershot.

We can now understand how it happens that Galileo, Mersennus, and others, could compare the impulse

given by a falling body with the pressure of a weight Impulsion. in the opposite scale of a balance, and can see the reason of the immense differences, yet accompanied by a sort of regularity, in the results of the experiments. Galileo, Mersennus, and Riccioli, found them to be proofs that the forces of moving bodies are as their velocities; because the heights from which the body fell were as the squares of the weights started from the ground. Gravefande found the same thing as long as he held the same opinion; but when he adopted the Leibnizian measure, he found many faults in the apparatus employed in his former illustrations, and altered it, till he obtained results agreeable to his new creed. But any one who examines with attention all that passes in the bending of the apparatus, and takes into account the mass of matter which must be displaced before the opposite arm rises so far as to detach the spring which gives indication of the magnitude of the stroke, must see that the agreement is purely accidental, and may be procured for any theory we please (see Gravefande's Nat. Phil. translated by Desaguliers, vol. i. p. 241. &c.). The proposition, n° 95, DYNAMICS, suffices for explaining every thing that can happen in such experiments. And it will shew us, that although the motion of impulse is produced by pressure alone, yet impulse is incomparable with mere pressure: It is not infinitely greater, but disparate. A weight (which is a pressure) bends a spring to a certain degree, and will derange to a certain degree the fibres of a body on which it presses, before it be balanced. The same weight, falling on this spring from the smallest height, will bend it farther, and may crush or shiver to pieces the body which would have carried it for ever. We shall make some further remarks on this subject, of great practical importance, under the word PERCUSSION.

⁷⁸ THE method which we have pursued in considering CONVERS. the doctrines of impulse, differs considerably from that which has generally been followed; but we trust that it will not be found the less instructive. Although the reader should not adopt our decided opinion, that we have no proof of pure impulse ever being observed, and that all the phenomena which go by that name are really the effects of pressures, analogous to gravity, he perceives that our opinion does not lead to any general laws of impulse that are different from those which are acknowledged by all. We differ only, by exhibiting the internal procedure by which they are unquestionably produced in a vast number of cases, and which takes place in all that we have seen, in some degree. Our method has undoubtedly this advantage; that it requires no principle but one, namely, that accelerating forces are to be estimated by the acceleration which they produce. Even this may be considered, not as a principle, but merely as a definition. We get rid of all the obscurity and perplexity that result from the introduction of inertia, considered as a power—a power of doing nothing—and we are freed from the unphilosophical fiction (adopted by all the abettors of that doctrine, and even by many others) of conceiving the space, in which motions are performed, and bodies act, to be carried along with the bodies in it.—This furnishes, indeed, in some cases, a familiar way of conceiving the thing, by supposing the experiments to be made in a ship under sail, and by appeal-

impulsion. ing to the fact, that all our experiments are made on the surface of a globe that is moving with a very great velocity. But it is an absurdity in philosophy, and, when minutely or argumentatively used, it does not free us from one complication of action; for, before we can make use of this substitution, we must demonstrate, that the actions depend on the relative motions only: And this, when demonstrated, obliges us to measure forces by the velocity which they produce.

As no part of mechanical philosophy has been so much debated about as impulsion, it will surely be agreeable to our readers to have a notice of the different treatises which have been published on the subject:

Galilei Opera, T. I. 957. II. 479, &c.
Jo. Wallilii Tractatus de Percussion. Oxon. 1669.
Chr. Hugenius de Motu Corporum ex Percussion. Op. II. 73.
Traité de la Percussion des Corps, par Mariotte, Op. I. 1.
Hypothesis Physica Nova, qua phenomenonum causa ab unico quodam universali motu in nostro globo supposito repetuntur. Aufst. G. G. Leibnitzio. Moguntiae 1671.—Leibn. Op. T. II. p. II. 3.
Eiusdem Theoria Motus Abstracti. Ibid. 35.
Hermannii Phoronomia. Amst. 1716.

Discours sur les Loix de la Communication de Mouvement, par Jean Bernoulli, Paris, 1727. Jo. Bern. Oper. III.

Dynamique de D'Alembert.
Euleri Theoria Corporum solidorum seu rigidorum, 1765.

Borelli (Alphons) de Percussion.
See also M'Laurin's Fluxions, and his Account of Newton's Philosophy, for his Dissertation crowned by the Acad. des Sciences at Paris.—Also Dr. Jurin's elaborate dissertations in the Phil. Trans. N^o 479.—Also Gravefande's Nat. Philosophy, where there is a most laborious collection of experiments and reasonings; all of which receive a complete explanation by the 39th Prop. Princip. Neutoni I. or our n^o 95. DYNAMICS. There are also many very acute philosophical observations in Lambert's Gedanken über die Grundlehren des Gleichgewichts, und der Bewegung, in the second part of his Gebrauch der Mathematik.—Also, in the works of Kaestner, Hamburger, and Busch. Muschenbroeck also treats the subject at great length, but not very judiciously. We do not know any work which treats it with such perspicuous brevity as M'Laurin's Account of Newton's Philosophy.

Page 108. col. 1. Dele lines 24, 25, 26, 27, 28, 29; and in their place read, "which though Lord Auchinleck and his son took the same side, they took it with very different degrees of ardour. The judge saw not the propriety of illuminating his windows, when the cause was finally decided by the House of Peers; and to compel him to illuminate, the advocate got possession of a Chinese gong."
Page 186. col. 1. line 18. For "Henry VI." read "Henry II."
253. col. 1. line 23. for sulphuret read sulphat.
CHEMISTRY-Index, page 399. col. 1. line 2. After 464 add, "and Part 3. chap. 2. sect. 11. Page 399. col. 1. line 38. For p. 624, read 359.—Page 402, col. 2. line 18. For p. 624, read p. 359.
Page 451. col. 1. line 34. For "Diderot," read "D'Alembert."

Edinburgh: