16a 
MINUTES OP PBOCEEDINGS OP 
a simple yet striking instance of the great resistance which a very rare 
medium would offer to a solid moving in it with a high velocity. Thus 
Robins 1 states that Dr Halley thought it reasonable to believe that the 
opposition of the air to large metal shot is scarcely discernible, although in 
small and light shot, he acknowledged that it ought and must be accounted 
for. Robins further states, 2 that a musket-ball, fired with a charge of half 
its own weight of powder, w T ould leave the gun with a velocity of 1700 f.s., 
and that, with an elevation of 45°, its range would be 17 miles in a 
vacuum; whilst practical writers on the subject say that the range in air is 
short of half a mile. This resistance of the air p, depends upon the 
velocity v } upon the form, and upon the size of the moving body. When a 
body is at rest in air, the horizontal pressure tending to move it one way is 
just equal to the pressure tending to move it in the opposite direction. If 
the body be put in motion by any ^force, the pressure of the air tending to 
prevent motion is greater than it was before in that direction, and the 
• pressure in the direction of the motion is less than it was before. The 
difference of the pressures of the air before and behind the body is called 
the resistance of the air to that particular form of solid moving with the 
given velocity : but as the pressure of the air in direction of motion 
decreases rapidly as the velocity increases, it is commonly neglected. 
Thus, p is some function of the velocity v 3 as f{v), such that when the 
body is at rest, or v — 0, we have p =/(v) = 0. It is usual to assume 
that p can be expanded in a series of the form 
p = av + bv 2, + cw 3 + dv 4 + ev^ + &c. 
where a 3 h, c, &c. are to be determined by experiment, and are dependent on 
the form and size of the moving body. 
Newton’s experiments gave p = bv^ 3 and a = c = d = &c. = 0, 
Hutton’s gave. p — av + bv 2 , and .*. c = d = &c. = 0, 
Didion’s gave. p = bv 2 + cu 3 , and .*. a = d = &c. = 0, 
Colonel Mayevski’s gave... p = bv 2 + dv 4 , and .*. a = c = &c. = 0. 
It is possible that the resistance of the air, for a limited range of variation 
of velocity, may be tolerably represented by one or more of the above formulae, 
but we must be careful not to assume that these formulae must apply to all 
velocities of the same body, or that the same law will hold for all forms of 
bodies. Careful experiments must be made so as to discover some empirical 
law connecting the resistances of the air to bodies of simple forms and the 
velocities with which they move. When this is well done, all that is needed 
for practical use will have been obtained. But, beyond this, the mathe¬ 
matician will have been provided with facts which may serve as tests of the 
soundness of his theoretical investigations, and in the end he may succeed in 
forming a theory of resistances on mathematical principles, and thus replace 
our empirical formulae with one that is true and general. 
1 Gunnery. Preface, p. 48. 
2 p. 145. 
