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XIV. On the Resistance of the Air to the Motion of Elongated Projectiles having variously 
formed Heads. By F. Bashforth, B.l)., Professor of Applied Mathematics to the 
Advanced Class of Artillery Officers , Woolwich , and late Fellow of St. John's 
College , Cambridge. Communicated by Professor Stokes, Sec. B.S. 
Received January 30,— Read February 20, 1868. 
The famous theory of the parabolic motion of projectiles was at an early period found 
to give results not in accordance with practice. Manifestly, then, the air must offer a 
very sensible resistance to a body which is moving through it with a high velocity. 
This resistance will depend upon the form of the moving body, and upon the velocity 
with which it is moving. Hence, before the path of a projectile can be calculated, it 
will be necessary to determine experimentally the resistance opposed by the air to the 
motion of the projectile, corresponding to various velocities. According to Newton’s 
law, the resistance of the air varies as the square of the velocity. But the velocities 
were low in the experiments made under his direction. In 1719 John Bernoulli 
gave equations for finding by the method of Quadratures the path &c. of a projectile, 
when the resistance of the air was supposed to vary according to any power of the 
velocity. But in spite of grave doubts respecting the accuracy of Newton’s law, it 
has been adopted by most of the eminent mathematicians who have written on the sub- 
ject, such as Euler (1753), Lambert (1765), Borda (1769), Bezout (1789), Tempelhof 
(1788-9), d’Ehrenmalm (1788), Lombard (1796), and Poisson. 
The first good experiments made with a view to determine the resistance of the air 
to the motion of projectiles were those of Robins in 1742. The projectiles used were 
leaden bullets of small size. When we consider the great density of the material used, 
its liability to change its form in the barrel of the gun, and the smallness of the solid 
projectiles, it is truly wonderful that Robins was able to accomplish so much with his 
ballistic pendulum. Afterwards Hutton carried on Robins’ system of experimenting 
both with the whirling machine and ballistic pendulum, introducing additional pre- 
cautions, and using iron projectiles of greater size. In recent times MM. Didion, Morin, 
and Piobert have carried on experiments in France with heavier spherical projectiles, 
by the help of an improved ballistic pendulum ; but they have done little more than 
confirm the results of Robins and Hutton, and extend them to spherical projectiles of 
larger diameter. 
Robins came to the conclusions : — “ First , That, till the velocity of the projectile sur- 
passes that of 1100 feet in a second, the resistance may be esteemed to be in the dupli- 
cate proportion of the velocity ; and its mean quantity may be taken to be nearly the 
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