The foregoing presupposes that the air offers no resistance to the passage of the projectile through it. The fact however is quite otherwise, for no sooner does the projectile begin its flight than its velocity is constantly diminished by the air’s resistance. Now this resistance is complex, depending upon a number of different conditions, the effect of which can be taken into account only by extremely complex calculations. Obviously it will vary according to the area of the section presented by the projectile to the line of its flight, and again by the shape of its front, for a pointed shot will cleave the air with less resistance than one with a flat front. Then the density of the air at the time will also enter into the calculation. The mass of the projectile and also its velocity, upon which depend its _vis viva_, energy, or power of overcoming resistance in doing work, will also have to be considered. Most complex of all is the law, or rather laws (_i.e._ relations), which connect the air resistance with the velocity; for this relation no definite expression has been found. It is a function of the velocity (known only by experiment under defined conditions), and varying with the velocity itself. Thus for velocities up to 790 ft. per second, it is a function (determined experimentally) of the second power or square of the velocity; between 790 ft. per second and 990 ft. per second the law of resistance is changed and becomes a function of the third power of the velocity; between 990 ft. and 1,120 ft. velocity the law again changes and is related to the sixth power of the velocity; between 1,120 ft. and 1,330 ft. the resistance is again related to the third power of the velocity; and with higher speeds than that last named it is again more nearly related to the square of the velocity. It will be seen that to calculate the path of a projectile is really a very difficult mathematical problem, and indeed one which can be solved only approximately when all the known data are supplied. The air resistance to the motion of a projectile is much greater than before trial would be supposed. Let us take an experiment that has actually been recorded, in which a bullet three-quarters of an inch in diameter, weighing one-twelfth of a pound, was found to have a velocity of 1,670 ft. per second at a distance of 25 ft. from the gun, and this 50 ft. farther was reduced to 1,550 ft. per second. Now if the reader will calculate, according to the formula we have given above, the _energy_ due to the bullet’s velocity at these points, he will find it must have done 500 foot-lbs. units of work in traversing the 50 ft., and as this could have been expended only in overcoming the resistance of the air, we learn that this last must have been equivalent to a mean or average pressure of 10 lbs. thrusting the bullet backwards.