Si/stems of Physical Units. 85 



Of the four units, any two are chosen at pleasure. If we 

 choose for units of length and time the metre and the second 

 respectively, then the unit of velocity becomes the velocity of 

 a uniform 'motion in which 1 metre is described in 1 second; 

 and if the velocity of uniformly accelerated motion change by 

 one unit of velocity in one second, then its acceleration is 

 unity. But if we arbitrarily choose the second and the velo- 

 city "of light as units, then, according to equation (8), the unit 

 of length is the distance (40,000 geographical miles) which 

 light traverses in one second. The corresponding acceleration 



., 40000x7420 . , ,, 



will be, as we easily see, g^ times as great as the 



acceleration due to gravity. We need not follow out the two 

 other possible combinations. 



By choosing suitable units of mass and force we may elimi- 

 nate the constants of equation (6), which assumes the form 



k=ma (12) 



The six quantities — space, time, velocity, acceleration, force, 

 and mass — are connected by the four equations (2), (3), (6), 

 and (7); if we wish to eliminate the four constants, we must 

 retain the four units already determined. Equation (7) 

 assumes then the form 



*=p (13) 



The units of mass and force must be chosen so that, on the 

 one hand (equation 12), unit force communicates unit accele- 

 ration to unit mass, and so that, on the other hand, unit mass 

 attracts an equal mass at unit distance with unit force. Of 

 the six units any two may be chosen at pleasure. But if we 

 wish to choose three units at pleasure — say those of length, 

 time, and mass — then we must retain one constant, for ex- 

 ample that of equation (7) (the "constant of attraction "); 

 for, since the units have been some of them chosen at pleasure, 

 and some of them determined by means of the three remaining 

 equations, the constant of the equation in question does not 

 usually become unity. AYe may then regard this constant as 

 a quantity denned by the equation 



c=H (14) 



If we wish to choose four units at pleasure, say those of 

 length, time, force, and mass, we must retain one more constant, 

 for example that of equation (6). 



By the introduction of absolute units the equations of defi- 



