OF THE MOTION OF A SYSTEM. 203 



^) A ~ }—lm [ Pidx + Adx) + Q (d2/ + Ady) +E(d2 + 



Adz)|. 



The sum or integral of this expression, considered with 

 regard to the finite differences, may be denoted by 2^, the 

 sum of the similar expressions, derived from the separate 

 bodies of the system, being still distinguished by 2. Now 

 2^ m P(dx4-Adjr) is evidently equal toJmPdx: and we 



have =^m jf + 2, ^m { (a ^) + (a jf ) 



+ (a •^)* } — 22 /m(Pdx + Qdy + Rdz) : [for, if Aw be the 



finite difference of u, A (u^):=z(u + Auf — u^=:2uAu + Am-, 

 and A(i/2) + Au^=z 2mAm+2Am^ consequently u^ + l>, Au^ 

 =22^ (uAu + Au% and, in the present case dx^-\-^^ {Adxf 

 zz22^ (dx + Adx) Adx: and with respect to the integral of 

 mP {dx + Adx) it is evident that the expression being only 

 of one dimension, the product mPdx will remain unaltered, 

 whether it be supposed to vary by finite or by infinitely 

 small differences, provided that the same value of P be 

 always attributed to the same value of x, so that the dif- 

 ference of the values ofJmPdx for any two values of x 

 will be equal to the difference of the values of 2^m (Pd.r+ 

 Ada:) ; that fluent may, therefore, be considered as the 

 integral represented by the character 2^ .] If, therefore, 

 we denote by v, v\ v", . . . , the velocities of m, m', m", . . ., 



we shall have Imv^ = ^— ^z ^^ { (^ j^) + (a ^)^ + 



(A-j-Y + 2Xfm{Pdx+ Qdy + Rdx). Now the quantity 



under the sign 2^ being necessarily positive, we see that 

 the impetus of the system is diminished by the mutual 



