Nature and their Mutual Dependence. 7 



When Xdx -f Ydy -f Zdz is an exact differential which we may 

 denote by d.~F(x } y } z), then formula (5) by integration simply 

 gives m 



q = m .F(x,y,z)---.v* + C. ' • • ( 7 ) 



As a particular case, I will here regard that in which the 

 resistance P in the direction of the path is the constant, as in 

 Coulomb's experiments with friction of metal sliding on metal ; 

 agreeably to formula (6) we then get 



q = ?.s, 



when q is supposed = zero for s = 0. 



This equation shows that the newly developed energy is equal 

 to the product of the friction and the space passed through, 

 which is in accordance with my earlier experiments, and that 

 the amount of this energy is independent of the velocity with 

 which the slider is moved ; and this result was likewise derived 

 from my experiments. 



The Energy in a whole System of Material Points. 



Let us next advert to the motion of a whole system of material 

 points whose masses we may denote by m, m\ m", &c. 



After the lapse of time t, let x, y, z ; x\ y', z r ; x 11 , y n , z", &c. 

 be the coordinates of the points m, m\ m", &c, the accelerating 

 forces in direction of the axes for these points respectively 

 X, Y, Z ; .X', Y', Z'; X", Y", Z", &c, and let the increments of 

 the energies which are yielded by these points to the material 

 resistances be respectively dq, dq', dq" } &c. ; then, in consequence 

 of formula (5), we get 



*— [( x -50* + ( T - S> + ( z - £)*} 



"V'=-"[(x"-5>" + (y»- f->" + (z"- 5>»} 



&c. 



(3), which had then first to be subtracted from dQ in order to show the in- 

 crement of the lost energy, or the energy dq appearing in a new shape : 

 partly I have wished thereby to avoid the confounding of material resist- 

 ances with real forces ; for it seems to me that the material resistances are, 

 as it were, " a lifeless thing," to which some part of real force, being the re- 

 sultant of the three forces X, Y, Z in formula (1) resolved in direction of 

 the path, is imparted during the motion of the mass m. Though it is sure 

 that formula (6' ) may be regarded as a simple result of formula (1), yet I 

 keep this formula so much the more, that the train of ideas developed 

 above made me at first sensible of the real state of the whole. 



