(1.) Conservation of Momentum.-What is understood by this is a mere direct consequence of Newton's first interpretation of his third law of motion, viz., that Action and Reaction are equal and opposite. Stated in its simplest form it asserts that the momentum of a system of bodies, measured in any direction whatever, is not altered by their mutual action, whether that action be of the nature of traction, attraction, repulsion, or impact. And we see at once from this third law of motion that it must be so, because the change of momentum, in any direction, of any one part of the system, per unit of time, is the measure of the force acting on that part in that direction. Whatever momentum in this particular direction is gained by one member of the system must have been lost by other members, but not from their whole momentum, merely from the part of it in this direction. It thus appears that the (algebraic) sum of the momenta generated by the mutual actions of the system is zero.

These momenta are in fact directed magnitudes (like the forces of which they are the measure), and are therefore capable of cancelling one another. In this sense the conservation is of the same nature as that of the imagined electric or magnetic fluids, where no portion whatever of one kind can be produced without the simultaneous appearance of an equal quantity of the other, a quantity just capable of neutralizing it. This is obviously not in any sense analogous to the Conservation of Matter of which we have just spoken.

(2.) Conservation of Moment of Momentum.-Here we deal with quantities of the order of the moments of forces about an axis, i.e. couples in Poinsot's sense. These also are directed magnitudes depending for their conservation

upon the first interpretation of Newton's third law, and therefore the same remarks apply to them as to the preceding.

(3.) Conservation of Vis Viva.--Vis viva is the old name for energy or the power of doing work. We now deal with quantities which cannot possess direction, because they are essentially products of pairs of quantities similarly directed, and are therefore all to be treated as of the same algebraic sign, or rather (to adopt the language of Sir W. R. Hamilton) as signless quantities. With such there can of course be no cancelling.

To make our meaning clear, let us consider upon what vis viva depends. It depends upon and is proportional to the product of the mass into the square of the velocity. Now mass is of course a signless quantity; evidently we cannot have negative mass. Then with regard to the square of the velocity, this will be positive whether the velocity be positive or negative, whether it be in one direction or the opposite. Vis viva, therefore, or energy, is something which is not affected with the sign of direction, or, as we have already said, it is a signless quantity.

98. We have said that the energy which a body contains its vis viva-its power of doing work, is independent of the direction in which it is moving; and, further, that while the mass is the same, it is proportional to the square of the velocity. For instance, we may measure the energy of a cannon ball or of an arrow by the distance it will carry itself up against the force of gravity, represented by its own weight, when shot vertically upwards, and we find that with a double velocity it will go four times as high. Or we may point the cannon horizontally, and measure the energy of the same ball by the number of planks of oak wood which it can penetrate, and

we shall find that a ball with double the velocity will penetrate nearly four times as many as one with the single velocity. All these experiments concur together in convincing us that the energy of the ball is independent of the direction in which the cannon is pointed, and is proportional to the square of the velocity, so that a double velocity will give a fourfold energy.

99. We have just now spoken about a cannon ball fired into the air against the force of gravity. Such a ball, as it mounts, will each moment lose part of its velocity, until it finally comes to a standstill, after which it will begin to descend. When it is just turning it is perfectly harmless, and if we were standing on the top of a cliff to which it had just reached, we might without danger catch it in our arms and lodge it on the cliff. Its energy has apparently disappeared. Let us, however, see whether this is really true or not. It was fired up at us, let us say, by a foe at the bottom of the cliff, and the thought occurs to us to drop it down upon him again, which we do with great success, for he is smashed to pieces by the ball.

In truth, dynamics informs us that such a ball will again strike the ground with a velocity, and therefore with an energy precisely equal to that with which it was originally projected upwards. Now, when at the top of the cliff, if it had not the energy due to actual motion, it had nevertheless some sort of energy due to its elevated position, for it had obviously the power of doing work. We thus recognise two forms of energy which change into one another, the one due to actual motion and the other to position; the former of these is generally called kinetic, and the latter potential energy. All this appears to have been clearly perceived by Newton, who gave it as a second interpreta

tion of his Third Law of Motion. His statement is in language equivalent to the following:-Work done on any system of bodies has its equivalent in the form of work done against friction, molecular forces, or gravity, if there be no acceleration; but if there be acceleration, part of the work is expended in overcoming resistance to acceleration, and the additional kinetic energy developed is equivalent to the work so spent.

100. Thus Newton expressly tells us (though not in these words) that we are to include in the same category work done by or against a force-whether that force be due to gravity, friction, or molecular action (such as elasticity, for instance), or even to acceleration.

(a.) When work is done against gravity, as in lifting a mass from the ground, we have just seen that it is (as it were) stored up in the raised mass; we can recover it at any time by letting the mass descend. Thus it is that we furnish a clock with motive power sufficient to keep it going for a week in spite of friction and other resistance, by simply winding up its weights.

(b.) When work is done against molecular forces, we have a similar storing up, as, for instance, in drawing a bow or in winding up a watch.

(c.) When work is done against the inertia of a body, i.e. to accelerate its velocity, Newton's definitions show that the kinetic energy so produced is equal to the work so spent.

(d.) In abstract dynamics we simply consider as lost the work spent against friction. In Newton's time it was not known what became of it.

101. Leaving out then, for the present, the fourth alterna

1 See Thomson and Tait's Natural Philosophy, § 269; or Tait's Thermodynamics, § 91.

tive, we see that whatever work is spent we must, according to Newton, even in abstract dynamics recognise that it is not lost, but only transformed into an equivalent quantity stored up for future use, either in a quiescent form (as, for instance, a raised weight or bent spring), or in an active form (as vis viva of a moving mass). Here, then, at last, we recognise the same sort of conservation as that which we found in matter. But the statement so far is defective, as we have seen, in one particular. What becomes of work spent in overcoming friction? or what becomes of the energy of the blacksmith's hammer after it has struck the anvil? To this experiment alone can give the answer. Let us see

what it has told us.

Man has been called a reasoning animal, a laughing animal, according to the momentary whim or humour of the classifier; but he is perhaps still more definitely separated from all other animals when specified as the "cooking animal." Now, it has always appeared to us as something little short of marvellous, that even for the high purpose of cooking his food, or of inflicting exquisite torture on a vanquished foe, savage man should ever have hit upon the process of procuring fire by friction. Considering his condition, and comparing his opportunities and his success with those of even our greatest modern physicists, we cannot but look upon this as one of the very greatest and most notable discoveries ever made in physics. All the more notable, too, from the fact that a man like Newton, though of course aware of it, absolutely missed its significance even at the very moment when it alone was wanted to fill a serious lacuna in one of his grandest and most important practical generalisations. The missing link was all but supplied by Rumford and Davy at