in an irregular manner, and without any precautions for the safety of the workmen: the pickaxe, however, alone is used. The explosion by gunpowder would certainly bring down the roofs of the quarries, which are no where supported by props or bulwarks. The various specimens I collected here were:

1st. A greenish steatite, the surface of which, as well as its interior texture, is penetrated with a kind of pyritous varnish of a bronze colour, but so light and so efflorescent (if I may use the expression) that it seems as if the heart of the steatite, which is black, was shown through this kind of varnish. This pyritous steatite is very heavy; it blackens paper, and moves the magnetic needle strongly. It contains a small proportion of copper, but scarcely percep

tible.

2d. The same steatite, still richer in pyrites, with the slight yellowish or bronze varnish I have mentioned, which seems to gild the black serpentine rock, is soft to the touch, blackening the fingers, and forcibly obedient to the magnet. We find in the same rock arsenical magnetic pyrites, very heavy, and with a metallic fracture of a grayish white.

3d. This is a rare and superb specimen, being five inches nine lines long by four inches broad: the base of it is a serpentine of a deep black, a little glossy, blackening paper, without a pyritous appearance; but very heavy, and strongly attractable by the magnet, remarkable from needles of transparent white arragonite, one crystal of which is two inches three lines long and four lines in diameter, of a hexagonal figure, but always without a pyramid. Other crystals of a still greater diameter are to be seen, sometimes in kinds of cavities in the specimen, sometimes in the mass itself of the pyritous serpentine, and seem to have been formed simultaneously with the pyritous and magnesian elements of which this rock is composed.

4th. We find a few yards from the quarries, and a little lower down, some old excavations, but not so deep here there is a striated and silky-like pyrites, with beautiful green efflorescences of carbonated copper. This steatite is

soft to the touch, and has yellow ochrey spots in it, appa▾ rently proceeding from grains of altered cupreous and fer. ruginous pyrites. M. Ansaldo informed me, that this pyrites was formerly wrought for the sake of its sulphate of copper, but abandoned on account of its poverty.

LIX. Essay upon Machines in General. By M. Carnot, Member of the French Institute, &c. &c.

[Continued from p. 221.
PROBLEM.

XX. THE virtual movement being known of any given system of hard bodies, (i. e. that which it would assume if each of the bodies were free,) to find the real movement which it should have the following instant.

Solution. Let us denominate each molecule of the system,

Its virtual given velocity,

W

Its real velocity sought,

V

The velocity it loses, in such a manner that W is the result of V and of this velocity,

U

Let us now imagine that we make the system assume an arbitrary geometrical movement, and let the velocity which m will then have be

น

X

Y

Z

y

The angle formed by the directions of W and V, The angle formed by the directions of W and U, The angle formed by the directions of V and U, The angle formed by the directions of W and u, The angle formed by the directions of V and u, The angle formed by the directions of U and u, This being done, we shall have the equation smuƯ cosine x = 0 (F); by means of which we shall find in all cases the state of the system, by attributing successively to the indeterminates a different relations and arbitrary direc tions.

DEFINITIONS.

XXI. Let us imagine a system of bodies in movement in

any

any given manner : let m be the mass of each of these bodies, and V its velocity; let us now suppose that we make the system assume any geometrical movement, and let u be the velocity which m will then have, (and what I shall call its geometrical velocity,) and let y be the angle comprehended between the directions of V and u; this being done, the quantity mu V cosine y will be named the momentum of the quantity of movement mV, with respect to the geometrical velocity u; and the sum of all these quantities, namely smu V cosine y, will be called the momentum of the quantity of movement of the system with respect to the geometrical movement which we have made it assume: thus the momentum of the quantity of movement of a system of bodies, with respect to any geometrical movement, is the sum of the products of the quantities of movement of the bodies which compose it, multiplied each by the geometrical velocity of this body, estimated in the ratio of this quantity of movement. In such a manner that by preserving the denominations of the problem, smu W cosine x is the momentum of the quantity of movement of the system before the shock; s m u V cosine y is the momentum of the quantity of movement of the same system after the shock; and s m u U cosine z is the momentum of the quantity of movement lost in the shock (all these momenta being referred to the same geometrical movement). Thus, from the fundamental equation (F) we may conclude, that in the shock of hard bodies, whether these bodies be all moveable, or some of them fixed, or, what comes to the same thing, whether the shock be immediate, or made by means of any machine without spring, the momentum of the quantity of movement lost by the general system is equal to zero.

W being the result of V and U, it is clear that we have W cosine x cosine y + U cosine z, or mu W cosine x = m u V cosine y+m u U cosine z, or lastly, 8 m u W cosine x = smu V cosine y + s m u U cosine ≈ : now we have found s m u U cosine z = 0; therefore smu W cosine x + smu V cosine y, that is to say, in respect to any geometrical movement, the momentum of

U 4

the

the quantity of movement of the system, immediately after the shock, is equal to the momentum of the quantity of movement immediately before the shock.

When we decompose the velocity which a body would assume if it were free, into two, one of which is the velocity it actually assumes, and the other the velocity it loses; and reciprocally if we decompose the velocity it loses, into two, one of them being that which it would have taken if it had been free, the other will be the velocity it gains: whence it visibly follows, that what we understand by the velocity gained by a body, and what we understand by its velocity lost, are two quantities equal and directly opposite: this being done, the momentum of the quantity of movement lost by m, with respect to the geometrical velocity u, being, according to the preceding definition, m u U cosine z, the momentum of the quantity of movement gained by the same body will be mu U cosine z; for there is no difference between these two quantities, except in this, that the angle comprehended between u and the velocity gained is the supplement of that comprehended between u and U; so that one of these angles being sharp, the other will be obtuse, and its cosine equal to the cosine of the other, taken negatively.

Hence it follows, that the momentum of the quantity of movement lost by the general system, with respect to any geometrical movement, (which is null, as we have seen above,) is the same thing as the difference between the momentum of the quantity of movement lost by any part of the bodies which compose it, and the momentum of the quantity of movement gained by the other bodies of the same system thus this difference is equal to zero, and thus the one of these two quantities is equal to the other; that is to say, the momentum of the quantity of movement lost in the shock by any part of the bodies of the system, with respect to any geometrical movement, is equal to the momentum of the quantity of movement gained by the other bodies of the same system.

We may, therefore, from the preceding definition, colect the three propositions contained in the following

THEOREM.

THEOREM.

XXI. In the shock of hard bodies, whether this shock be immediate, or whether it be made by means of any machine without spring, it is clear that in respect to any geometrical movement,

1st. The momentum of the quantity of movement lost by the whole system is equal to zero.

2d. The momentum of the quantity of movement lost by any part of the bodies of the system, is equal to the momentum of the quantity of movement gained by the other part.

3d. The momentum of the quantity of real movement of the general system, immediately after the shock, is equal to the momentum of the quantity of movement of the same system, immediately before the shock.

It is clear, from the preceding definition, that these three propositions are radically the same, and are nothing else. than the same fundamental equation (F) expressed in different ways.

We may also remark that these propositions bear a great relation to those we extract from the consideration of the nomenta relatively to different axes; but the latter are less general, and are easily inferred from those established in XVII.

There is, therefore, as we see by the third proposition of this theorem, in every percussion or communication of movement, whether immediate, or caused by the intermedium of a machine, a quantity which is not altered by the shock: this quantity is not, as Descartes thought, the sum of the quantities of movement; nor is it the sum of the active forces, because the latter is only preserved in the case where the movement changes by insensible degrees, as we shall sec lower down, and it always diminishes when there is percussion, as will be proved in the second corollary. When the system is free, the quantity of movement estimated in any ratio, is in truth the same before and after the percussion; but this preservation does not take place if there are obstacles, any more than that of the momenta of quantity of movements referred to different axes: all these quan

tities