motion op projectiles — poisson 11 



uble because his illustrious countrymen had not solved it. Now 

 the numerical calculation of the integrals which express the time 

 and the two coordinates of the moving body, in functions of a 

 fourth variable, is effected as simply as the question allows, and 

 enables the approximations to be carried as far as we wish. We 

 can see an example in the "Exercises du calcul Integral" of Legen- 

 dre 6 in which these coordinates are calculated to within less than 

 a hundred-thousandth part of their values. 



Independently of the centrifugal force arising from the rotation 

 of the earth (which influences the motions of heavy bodies by 

 diminishing the force of gravity by a quantity that varies with 

 the latitude), this rotation also produces in these motions certain 

 deviations that it is interesting to understand, either in themselves 

 or in order to know to what extent they can influence the trajec- 

 tory of the projectiles, and whether it is necessary to consider them 

 in the practice of artillery. 



Many physicists have measured, with as much precision as has 

 been possible, the small distances by which bodies that fall from 

 a considerable height deviate from the foot of the vertical. La- 

 place and Gauss submitted this question to the calculus, but in 

 integrating the equations of this almost exactly vertical motion 

 they have left out of consideration the resistance of the air, which 

 can, however, sometimes have a very great influence on the result. 

 1 have therefore thought it would be useful to go over this problem 

 entirely and to extend the solution to the general case in which 

 the projectile is projected into the atmosphere with any velocity 

 and in any direction whatever. 



To this end I have in the first place formed the differential 

 equations of the absolute motion in space by referring the coordi- 

 nates of the moving body to fixed axes; then I have deduced from 

 these the equations of apparent motion such as we observe near 

 the surface of the earth, referred to fixed axes at the surface which 

 participates as well as we ourselves in the rotation of the earth. 

 These differential equations are very complicated, but by taking 

 the second of time for the unit of time, the angular velocity of 

 the diurnal motion becomes a very small fraction, which permits 

 (us) to reduce them to a more simple form. From these we deduce 

 some general consequences, enumerated as follows: 



(i) The motion of the earth prevents a liquid contained in a 

 vase and turning with a constant velocity about a vertical axis 



8 Vol. I, p. 33 6. 



