1884.] and change of units. 139 



hence may regard the equation (1) in the final general form of 

 the expression of any physical law. 



(2) We have already mentioned that the numerical value of 

 k depends upon the units [Q], [X], [Y], [Z]... employed to measure 

 the different quantities. We may therefore assign to h any value 

 we please by a suitable choice of any one of the units [Q], [X], [F]. . . 



For many treasons it is convenient that k should be unity, 

 and therefore the most usual assumption is that the unit of [Q] 

 should be so chosen that k shall be unity. 



In the same manner x, y and z can generally be connected by 

 physical laws with the three units of mass, space and time, and 

 we may thus obtain k = 1 for a large number of physical equations, 

 provided the whole series of units are chosen on the principle here 

 indicated. We thus see that systems of units can be formed based 

 on three fundamental units, such that a whole series of physical 

 laws, expressing relations between the quantities measured, can be 

 represented by ordinary equations with constant unity, instead of 

 by variation equations. We thus arrive at systems of units founded 

 on this principle, and a unit belonging to such a system is called 

 an absolute unit. For such a unit the right of arbitrary choice has 

 been given up, and it is agreed that the choice shall be directed by 

 a consideration that the quantity k in certain equations shall be 

 made equal to unity. 



It follows from this that when the three fundamental units are 

 selected the rest of the units belonging to the system are thereby 

 defined, and that if the fundamental units are altered, correspond- 

 ing alterations must take place in the whole system based upon 

 the three fundamental units, in order that the &'s may be still 

 maintained equal to unity. 



Let us consider the change from one system of absolute units 

 to another, both founded upon the same principle, that is to 

 say, both agreeing that the same &'s shall be unity. 



The equation between the numerical measures of q, x, y, z... 

 thus becomes for both systems 



q = x a yP z y ... . 

 Let the unit of x be changed from [X] to [X'], 



y m ... [F], 



* m ... w\ 



and in consequence 



q .- [Q] ■»[#]. 



Then if q, x , y', z be the new numerical measures of the same 

 actual quantities measured, we have 



