these passive forces from which nothing can be subtracted being always resisting, it is evident that the movement must continually slacken and from what we have said (XLV.), we see that if bodies are not solicited by any motrix force, the amount of the active forces will be reduced to nothing; i. e. the machine will be reduced to a state of rest, when the momentum of activity, produced by the friction since the commencement of the motion, will have become, equal to half the amount of the initial active forces: and if the bodies are heavy, the motion will finish when the momentum produced by the frictions shall be equal to half the amount of the initial active forces, plus the half of the active force which would take place if all the points of the system had one common velocity, equal to that which is owing to the height of the point where the centre of gravity was at the first instant of the motion, above the lowest point to which it can descend: this is evident from (XLII).

It is easy to apply the same reasoning to the case of springs, and in general to all cases in which the friction being subtracted, the soliciting forces are obliged, in order to make the machine pass from one position to another, to exercise a momentum of activity as great as that which is produced by the resisting forces when the machine returns from this last position to the former.

The motion would end much sooner if some percussion took place, since the sum of the active forces is always diminished in such cases (XXIII).

It is therefore evident, that we ought entirely to despair of producing what is called a perpetual motion, if it be true that all the moving powers which exist in nature are nothing else than attractions, and that this force, as it should seem, has a general property, that of being always the same at equal distances between given bodies, i. e. of being a function which only varies in cases where the distance of these bodies itself varies.

LXII. One general observation resulting from all that has been said, is, that the kind of quantity to which I have given the name of momentum of activity, performs a very conspicuous part in the theory of machines in a state of motion

for

for it is in general this quantity which we must economize as much as possible, in order to draw all the effect we can from one agent.

If it be required to raise a weight, water for example, to a given height; you will be able to raise more in a given time, not from having exhausted a greater quantity of power, but in proportion as you have exercised a greater momentum of activity (XLIV).

If it be required to turn a mill, either by water, or wind, or animals, it is not necessary that the shock of the water, the wind, or the effort of the animal be greater; but these agents should be made to consume the greatest momentum of activity possible.

If we wish to make a vacuum in the air in any way whatever, we must, in order to succeed, consume a montentum of activity as great as that which would be necessary for raising to the height of 30 feet a volume of water equal to the vacuum which we wish to produce.

If it be a vacuum in an indefinite mass of water like the sea, we must consume the same momentum of activity as if the sea were a vacuum; as if the vacuum which we wish to make were a volume of sea water, and as if we must raise this volume to the height of the level of the sea.

If it be required to produce a vacuum in a vessel of a given figure, it is evident that we cannot succeed without causing to ascend the centre of gravity of the total mass of the fluid in a quantity determined by the figure of the vessel; we must therefore consuine a momentum of activity equal to that which would be necessary to raise all the water in the vessel in a quantity equal to that from which the centre of gravity of the fluid must ascend.

In a machine at rest, where there is no other force to overcome except the vis inertia of the bodies, if we wish to produce any movement by insensible degrees, the momentum of activity which we have to consume will be equal to half the amount of the active forces we wish to produce; and if it be merely required to change the movement it has already, the momentum of activity to be produced will only be the

quantity

quantity in which this half amount will be increased by the change (XLV).

Finally, supposing we have any system of bodies, that these bodies attract each other, on account of any function of their distances; even supposing, if we please, that this law is not the same with respect to all the parts of the system, i. e. that this attraction follows any law we please, (providing that, between two given bodies, it only varies when the distance of these bodies in itself varies,) and it be required to make the system pass from any given position to another this being done, whatever be the path that we wish each of the bodies to take, in order to attain this object, whether we put all these bodies in motion at once, or the one after the other, whether we conduct them from one place to another by a rectilinear or curvilinear motion, and varied in any manner (providing no shock nor rapid change occur); lastly, whether we employ any kind of machines whatever, even by a spring, providing that in this case we ultimately replace the springs in the same state of tension in which they were at the first moment, the momentum of activity which they will have to consume, in order to produce this effect, the external agents employed to move this system, will always be the same, supposing the system to be at rest at the first instant of the movement, and at the last also.

And if, besides all this, it be necessary to produce in the system any given movement, or if it be already in motion at the first moment; and if it be requisite to modify or change this movement, the momentum of activity which the external agents will have to consume will be equal to that which it would be necessary to consume if it were merely requisite to change the position of the system, without impressing any motion upon it (i. e. considered as at rest at the first and last instants,) plus the half of the quantity by which we must augment the sum of the active forces.

It is of very little importance therefore, as to the expenditure or momentum of activity to be consumed, that the forces employed are great or small, that they employ such

and

and such machines, or that they act simultaneously or not: this momentum of activity is always equal to the produce of a certain force, by a velocity, and by a time, or the sum of several products of this nature; and this sum should always be the same, in whatever way we take it: the agents therefore will gain nothing on the one hand, which they do not lose on the other.

To conclude, let us suppose that in general we have any system of animated bodies, of any motrix forces, and that several external agents, such as men or animals, are employed to move this system in various and different ways, either by themselves or by machines :-This being granted,

Whatever be the change occasioned in the system, the momentum of activity consumed during any time by the external powers, will be always equal to the half of the quantity by which the sum of the active forces will have augmented during this time, in the system of bodies to which they are applied: minus the half of the quantity by which this same sum of active forces would have augmented, if each of the bodies were freely moved upon the curve it has described, supposing that it had then undergone at each point of this curve the same motrix force as that which it really undergoes providing always that the motion changes by insensible degrees, and that if we employ machines with springs, we leave these springs in the same state of tension in which we found them. [To be continued.]

LIV. Memoirs of the late ERASMUS DARWIN, M. D. [Continued from vol. xxx. p. 115.]

DARWINIANA.

HAVING laboured under a severe illness, the author of this memoir must apologize for so long delaying the continuation of the remarkable medical opinions of the great Dr. Darwin, whose powers of mind, fully bent upon one important subject, namely health, and the causes of disease, and the remedies to be applied, with the rationale of each, cannot fail to interest the philosophic world. U Dr.

Vol. 31. No. 124. Sept. 1808.

Dr. Darwin relates a remarkable cure of bleeding piles.Mrs. - had for twelve or fifteen years, at intervals of a year or less, a bleeding from the rectum without pain; which, however, stopped spontaneously after she became weakened, or by the use of injections of brandy and water. Lately the bleeding continued above two months, in the quantity of many ounces a day, till she became pale and feeble to an alarming degree. Injections of solutions of lead, of bark, and salt of steel, and of turpentine, with some internal astringents and opiates, were used in vain. An injection of the smoke of tobacco, with ten grains of opium mixed with the tobacco, was used, but without effect the two first times on account of the imperfection of the machine: on the third time it produced great sickness and vertigo, and nearly a fainting fit; from which time the blood entirely stopped. Was this owing to a fungous excrescence in the rectum; or to a blood-vessel being burst from the difficulty of the blood passing through the vena porta from some hepatic obstruction, and which had continued to bleed so long? Was it stopped at last by the fainting fit? or by the stimulus of the tobacco?

His method of curing spitting of blood is equally new and extraordinary.-Venous hæmoptoe frequently attends the beginning of the hereditary consumptions of dark-eyed people; and in others, whose lungs have too little irritability. These spittings of blood are generally in very small quantity, as a tea-spoonful; and return at first periodically, as about once a month; and are less dangerous in the female than in the male sex, as in the former they are often relieved by the natural periods of the menses. Many of these patients are attacked with this pulmonary hæmorrhage in their first sleep; because in feeble people the power of volition is necessary, besides that of irritation, to carry on respiration perfectly; but, as volition is suspended during sleep, a part of the blood is delayed in the vessels of the lungs, and in consequence effused, and the patient awakes from the disagreeable sensation.

M. M. Wake the patient every two or three hours by an alarum clock. Give half a grain of opium at going to bed,

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