A.—­The atmospheric resistance, no doubt, increases as the square of the velocity, and the power, therefore, necessary to overcome it will increase as the cube of the velocity, since in doubling the speed four times, the power must be expended in overcoming the atmospheric resistance in half the time.  At low speeds, the resistance does not increase very rapidly; but at high speeds, as the rapid increase in the atmospheric resistance causes the main resistance to be that arising from the atmosphere, the total resistance will vary nearly as the square of the velocity.  Thus the resistance of a train, including locomotive and tender, will, at 15 miles an hour, be about 9.3 lbs. per ton; at 30 miles an hour it will be 13.2 lbs. per ton; and at 60 miles an hour, 29 lbs. per ton.  If we suppose the same law of progression to continue up to 120 miles an hour, the resistance at that speed will be 92.2 lbs. per ton, and at 240 miles an hour the resistance will be 344.8 lbs. per ton.  Thus, in doubling the speed from 60 to 120 miles per hour, the resistance does not fall much short of being increased fourfold, and the same remark applies to the increase of the speed from 120 to 240 miles an hour.  These deductions and other deductions from Mr. Gooch’s experiments on the resistance of railway trains, are fully discussed by Mr. Clark, in his Treatise on railway machinery, who gives the following rule for ascertaining the resistance of a train, supposing the line to be in good order, and free from curves:—­To find the total resistance of the engine, tender, and train in pounds per ton, at any given speed.  Square the speed in miles per hour; divide it by 171, and add 8 to the quotient.  The result is the total resistance at the rails in lbs. per ton.

500._Q._—­How comes it, that the resistance of fluids increases as the square of the velocity, instead of the velocity simply?

A.—­Because the height necessary to generate the velocity with which the moving object strikes the fluid, or the fluid strikes the object, increases as the square of the velocity, and the resistance or the weight of a column of any fluid varies as the height.  A falling body, as has been already explained, to have acquired twice the velocity, must have fallen through four times the height; the velocity generated by a column of any fluid is equal to that acquired by a body falling through the height of the column; and it is therefore clear, that the pressure due to any given velocity must be as the square of that velocity, the pressure being in every case as twice the altitude of the column.  The work done, however, by a stream of air or other fluid in a given time, will vary as the cube of the velocity; for if the velocity of a stream of air be doubled, there will not only be four times the pressure exerted per square foot, but twice the quantity of air will be employed; and in windmills, accordingly, it is found, that the work done varies nearly as the cube of the velocity of the wind.  If, however, the work done by a given quantity of air moving at different speeds be considered, it will vary as the squares of the speeds.