Nature and their Mutual Dependence, 5 



direction of the path 



and the velocity in the path, after the lapse of time /, is put 

 equal to v, then we have the increment of energy expressed in 

 general by 



dQ = m .<fc .ds = m .<fi . vdt. 



We also perceive that the unit of this quantity Q is still the 

 same as observed before, viz. one pound raised one foot. 



But if the rectangular coordinates of the material point are 

 denoted by x, y, z, and the accelerating forces in the direction 

 of the three coordinate axes by X, Y, Z, then we have 



<£-X — + Y^+Z~, 

 ds ds ds 



which, substituted in the above equation, gives 



W-^tA+z*)*.* . . (i) 



from which the energy produced during the time t is found, viz. 

 Q=mj(X^+Y^ + Z^) + C iy .... (2) 



Cj being an arbitrary constant. 



If, on the contrary, the point is not perfectly free, but subject 

 to any material resistance, such as the resistance of a fluid, resist- 

 ance of friction, &c, then the increment of energy during the 

 time dt will only be 



7 / d*x dx d^ii dii d q z dz\ds lL r/ON 



dw — m\ -r^ - -r -f -~ • -~ -f -ns • -r ) tt • dt, . (6) 



\dt 2 ds d? ds dt~ ds J dt 3 w 



from which we find the energy, which in fact is contained in the 

 point after the lapse of time /, viz. 



v 2 

 w — m.~ +C 2 , (4) 



where C 2 is an arbitrary constant. 



The measure for this energy is still, as before, 1 pound raised 

 1 foot, which is easily ascertained by observing that the quantity 

 of energy w might also have been obtained by causing the mass 

 m to fall through a height h of vacuum so great that the terminal 

 velocity thereby had been v, which depth of fall is determined 

 from the equation 



v 1 



— —g ,h } as g is the force of gravity. 



