280 CELESTIAL MECHANICS. 1. vii. 32. 



three forces X, Y, and Z. These forces are simply de- 

 pendent on X, y, and z, when their intensity is invariable 

 in magnitude and direction ; but when they are directed to 

 moveable centres of attraction, or are dependent on the 

 mutual .action«i of the particles, their values will compre- 

 hend the time that has elapsed: so that calling the time if, 

 we may consider the forces X, Y, Z, in general as func- 

 tions of a:, y, z, and t. 



" Now if we call the velocity of the element, to which 

 the ordinates x, y, and z belong, reduced to the direction 

 of the axes, m, v, and w, these quantities will be unknown 

 functions of x, y, z, and t; they must depend on the ordi- 

 nates X, y, z, because, at the same instant, or for the same 

 value of ty the velocity may vary between one particle and 

 another in magnitude and in direction : they must also de- 

 pend on the time t, because in the same place, and for the 

 same original values of x, y, z, the velocity may change, 

 from one instant to another. If we wish to compare the 

 velocities of any one particle in two consecutive instants, 

 we must suppose that the variable quantity t becomes ^H- 

 d^, [or rather ^ + A^]; and in the same time the coordi- 

 nates of the particles x, y, and z, will become [x + wAf, 

 y+vAty aud z + wAt]; for in virtue of the velocities w, v,w, 

 the same particle which belonged to the coordinates x^y^z, 

 at the end of the time f, will correspond to x + uAt, y + vAt, 

 and z + wAt, at the end of the time t + At, It follows, then, 

 that in order to obtain the variation of the quantities u, v, 

 and w, with regard to the same particle at the different 

 instants, we must take the differences with regard to t, 

 and with regard to x, y, and z, considering uAt, vAt, and 

 wAt as the elementary variations of tJiese quantities. We 

 have therefore 



