Systems of Physical Units. 83 



velocity. But since - is proportional to a certain velocity 

 (viz. the final velocity), this equation may be better written 



«=4 (3) 



In order to extend these conceptions to every case, Ave must 

 often take account of infinitesimals. Thus, for example, to 

 extend the conception of velocity to the case of motion not 

 uniform, it must be defined by the differential equation 



u=c%. ....... (i) 



Acceleration, when not uniform, is defined by the equation 

 dli _ 



a = C oU <°) 



But even in these new forms the equations express the same 

 things as in their simpler form. Velocity is proportional to 

 the increase of distance, and acceleration to the increase of 

 velocity in the unit time. The conceptions of force (£) and 

 mass (m) are connected by the equation 



k = cma (6) 



This equation may be regarded as a definition of force, if mass 

 be taken as a conception previously defined — or, on the other 

 hand, as a definition of mass, if the conception of force has 

 been first defined in any way. Each additional conception is 

 defined by a separate equation. Moreover, these quantities 

 may be connected by equations distinct from the defining 

 equations which express natural laws. Thus, for example, the 

 ^Newtonian law of gravitation gives the equation connecting 

 length, mass, and force, 



k=Cjj, (?) 



where k denotes the force with which a mass m attracts an 

 equal mass at a distance L. 



2. We have thus a certain number (p) of distinct equations 

 in which q quantities occur. Each equation may contain a 

 constant. The units of the different quantities may be chosen 

 at pleasure ; the numerical value of the constant in an equa- 

 tion depends on the units of the quantities occurring in the 

 equation. If, for example, we determine the constant of equa- 

 tion (2) from the velocity of light, and take the geographical 

 mile as unit of length, the second as unit of time, and the 

 Ct2 



