its bearings.  The reason why of it is very simple.  Returning for a moment to our gun, we see that a certain amount of work is done on it in causing it to recoil; but the whole of the work done by the powder is, other things being equal, a constant quantity.  The sum of the work done on the shot and on the gun in causing their motions is equal to the energy expended by the powder, consequently the more work we do on the gun, the less is available for the shot.  It can be shown that, if the gun weighed no more than the shot, when the charge was ignited the gun and the shot would proceed in opposite directions at similar velocities—­very much less than that which the shot would have had had the gun been held fast, and very much greater than the gun would have had if its weight were, as is usually the case, much in excess of that of the shot.  In like manner, part of the work of a steam engine is done in driving the ship ahead, and part in pushing the water astern.  An increase in the weight of water is equivalent to an augmentation in the weight of our gun and its carriage—­of all that, in short, takes part in the recoil.

But, it will be urged, it is just the same thing to drive a large body of water astern at a slow speed as a small body at a high speed.  This is the favorite fallacy of the advocates of hydraulic propulsion.  The turbine or centrifugal pump put into the ship drives astern through the nozzles at each side a comparatively small body of water at a very high velocity.  In some early experiments we believe that a velocity of 88 ft. per second, or 60 miles an hour, was maintained.  A screw propeller operating with an enormously larger blade area than any pump can have, drives astern at very slow speed a vast weight of water at every revolution; therefore, unless it can be shown that the result is the same whether we use high speed and small quantities or low speed and large quantities, the case of the hydraulic propeller is hopeless.  But this cannot be done.  It is a fact, on the contrary, that the work wasted on the water increases in a very rapid ratio with its speed.  The work stored up in the moving water is expressed in foot pounds by the formula

  W v squared / 2g

where W stands for the weight of the water, and v for its velocity.  But the work stored in the water must have been derived from the engine; consequently the waste of engine power augments, not in the ratio of the speed of the water, but in the ratio of the square of its speed.  Thus if a screw sends 100 tons of water astern at a speed of 10 ft. per second per second, the work wasted will be 156 foot tons per second in round numbers.  If a hydraulic propeller sent 10 tons astern at 100 ft. per second per second, the work done on it would be 1,562 foot tons per second, or ten times as much.  But the reaction effort, or thrust on the ship, would be the same in both cases.  The waste of energy would, under such circumstances, be ten times as great with the hydraulic propeller