(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 of motion or the consequent 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. Compare, or rather contrast, this with the definition of momentum given above, and it will be seen that _vis viva_ is the product of the momentum and 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 of motion, is something which is not affected with the sign of direction, or, as we have already said, it is a signless quantity. It is found to be convenient to measure it as _half_ the product of the moving mass into the square of its velocity. So measured, it is now called (see § 99) _kinetic energy_. Now to our cannon again. Before firing there is no _vis viva_ of either cannon or ball. After firing each has _vis viva_, but that of the ball is greater than that of the cannon in the proportion in which the cannon’s mass exceeds that of the ball. And the system as a whole has _vis viva_ though it has no momentum. If, as before, we could reverse the motions of cannon and ball, then, even when they impinged, the _vis viva_ would _not_ be lost. As will presently be seen, it would be employed in heating both the impinging bodies.