AIR TO THE MOTION OF ELONGATED PROJECTILES. 
489 
Although the motion of a shot may be well represented by supposing a retarding 
force = — 2 bv* to act through a range of 1400 feet, there is reason to suppose that for 
velocities ranging from 1500 to 900 feet per second the value of b will be less for the lower 
velocities with equal degrees of steadiness. It unfortunately happens, however, that 
the angular velocity imparted to a shot, which most probably remains little changed 
during the time of flight, depends directly upon the initial velocity of the shot. Hence, 
when shot are fired with low initial velocities with a view to determine the value of b 
for low velocities, the steadiness of the shot is diminished, and therefore there is an 
increase of the resistance of the air on this account. The only way to meet the diffi- 
culty is to place screens near the gun to find the initial velocity, and others at a distance 
of 2000 yards or more, and so compare theory and experiment. 
It is worthy of notice that if a body move in a straight line under the action of a 
force varying as the velocity cubed, the mean velocity obtained by dividing space by 
time is exactly the actual velocity at the middle point of that space. Thus 
space 2s 2s 1 
time of describing space 2s 2as + 4bs 2 a + 2bs ' e 0C1 ^ a 1S ance s ' 
The date of the Report of the above experiments was October 23, 1866. 
I have long been aware that Major Otto had made trial of various laws of the resist- 
ance of the air in a work published in 1855. The law of the cube of the velocity was 
tried, but without any definite result*. It was in April 1867 that I first learnt that 
M. HELiEf had proposed the law of the cube of the velocity as the law of the resistance 
of the air to elongated projectiles in a work dated 1865, which law he had deduced 
from experiments made at Gavre in 1859, 1860, and 1861 J. It will be convenient to 
quote his own statement of the best series of experiments made at Gavre in 1859, in 
order to show the nature of the work done, and the perfect independence of my own 
methods and numerical results. M. Helie used one of the electro-ballistic pendulums 
to measure his velocities, but he does not state distinctly which it was. 
“Si la resistance de Fair est reellement proportionnelle au cube de la vitesse, on doit 
avoir un resultat sensiblement constant en substituant, dans l’expression v ~ lt ^ , les valeurs 
de v\ v" et x correspondantes a la meme charge. 
* Hilfsmittel fur ballistische Rechnungen, 1855, p. 12. t Traite de Balistique, 1865. 
t [In a Memoir, “ Etudes de Balistique experimentale,” presented to the Belgian Academy by Captain P. C. 
BoniENGh, June 12, 1867, the author, having deduced the cubic law of resistance of the air from his experiments, 
proceeds to remark : — 
“ Ce resultat est en accord complet avec les travaux les plus recents faits en France ; en effet, les experiences 
executees par la commission des principes du tir, en 1856 et 1857, ont conduit M. le capitaine Welter, pro- 
fesseur 4 l’Ecole duplication de l’artillerie et du genie de Metz, a recounaitre que la resistance de fair sur les 
projectiles spheriques est simplement proportionnelle au cube de la vitesse. 
“ Cette loi, admise depuis 1862, comme base des etudes balistiques ii cette ecole, a fourni des formules tres- 
simples et tres-facilement calculables sans l’intervention de tables, se pretant a des recherches que les anciennes 
forinules balistiques ne permettaient pas d’aborder, et donnant des resultats plus conformes a la pratique” (p. 84). 
—Aug. 1, 1868.] 
