MOTION OF PROJECTILES POISSON 9 



reduction is too great, and Borda has concluded from his own 

 observations that the measure of the resistance must be only 

 diminished to three-fifths of its theoretical value. From the theory 

 of Newton as modified by experiment, the retarding force relative 

 to the unit of mass for a sphere moving through the air has for 

 its expression the square of the velocity of the sphere divided by 

 its diameter and by the ratio of its density to that of the fluid, 

 and multiplied by a numerical coefficient concerning which the 

 writers on ballistics do not agree. According to Lombard, 2 and 

 relying on the experiments of Borda, this coefficient should be 

 equal to about nine-fortieths. But the true law of the resistance 

 as a function of the velocity is far more complex; for motions 

 which are either very rapid or very slow the coefficient seems 

 to deviate considerably from being proportional to the square of 

 the velocity; in the case of very great velocities it increases at 

 a much greater ratio, and on the contrary when it is a question 

 of small velocities, such as the very small vibrations of the seconds 

 pendulum 3 this coefficient is proportional to the simple velocity. 

 In order to determine directly and without any hypothesis the 

 law of the resistance that a body meets with in moving through 

 a fluid , it will be necessary to consider at the same time both the 

 motion of the body and that which the moving body communi- 

 cates to the fluid; as the result of this double motion the fluid 

 exerts at each instant a certain pressure at each point of the 

 moving body and normal to its surface; this pressure is different 

 from that which occurs in the state of rest and produces the resist- 

 ance, properly so-called, that the moving body experiences, and 

 to which it will be necessary also to add the force tangential to 

 the surface of the body arising from the friction of this body against 

 the layer of fluid in contact with it. In fact, this is what I have 

 been able to do in my Memoir on the simultaneous Motions of 

 the Pendulum and of the surrounding Air, 4 and which has led 

 me to deduce from theory the new correction which M. Bessel 

 has confirmed by experiment on the length of the seconds pendu- 

 lum. Hereafter I shall try to extend that analysis to the case of 

 the progressive motion of projectiles in the air and to determine, 

 if it is possible for me to do so, the pressure that the fluid dis- 

 placed by them exerts on their surfaces by its compression on one 

 side and expansion on the other, or the resistance that they 



2 Treatise on the motion of projectiles, p. 99. 



3 Additions to the Connaissance des Temps for the year 1834, p. 18. 



4 Memoirs of the Academy of Sciences, Vol. XI [Paris]. 



