6 M. L. A. Colding on the Universal Powers of 



v 2 

 If this value for - is substituted in the expression for w 3 formula 



(4), then we get w expressed simply in foot-pounds. 



The energy which the material point loses during the motion 

 in the time dt may consequently be represented by 

 dq = dQ—dw. 



But this energy is only apparently lost when it seems to vanish ; 

 for, so far as the stated principle is correct, it reappears in its 

 original magnitude merely under another form. Thus the new 

 energy may be represented by 

 7 r/ d 2 x\doo (^ T d?y\dy {„ d*z\dz~]ds 1± ,„. 



This equation, which may easily be put into the form 



dt, 



shows, to begin with, that we get the whole new energy by taking 

 the amounts of the energies which the accelerating forces in the 

 direction of the axes will separately produce. 



If we imagine the material point to be subject to any kind 

 of resistances, and if we denote the resultant of them all by K, 

 then this may be supposed to be resolved into two others — viz. 

 into the resistance in the direction of the path (which I shall 

 denote by P), and into the resistance perpendicular to the path 

 (which I shall call F ; ) . We have then, as known, 



■/ d*x\ dx / dSj\ dy ( d*z\ dzl 



\ A w)ds~ + V w) ds~ " r \r~d?)dsx 



which, substituted in the equation (5), gives 



<*?=*(§)"# 



from which we get by integration, 



ds 



f 



.■* + C, (6) 



in which C is an arbitrary constant. 



Hence follows that the newly produced energy depends only on 

 P, or the resistance in the direction of the path, whereas it is indepen- 

 dent of P p or the resistance perpendicular to the path*. 



* The last result may perhaps not appear to be essentially different from 

 what is immediately derived from formula (1), if we only look upon the ma- 

 terial resistances in the directions of the three coordinate axes as real forces, 

 which we might suppose to be included in the accelerating forces X, Y, 

 and Z ; but partly it is obvious that formula (1) would then represent the 

 increment of the quantity of energy which the moveable body in fact 

 would receive during the time t, consequently what is expressed in formula 



