Systems of Physical Units. 91 



given by the equation 



[L 1 ] = [M*N*] (30) 



Now we find 



«L=L 1 , 



[L]=4VL (31) 



where, again, n is the numerical value of a. If we introduce 

 into this equation the dimensions of [L x ] as well as the dimen- 

 sions of [L], if this unit is also a derived unit in the old 

 system, we obtain at any rate a relation into which the funda- 

 mental unit enters, which is to be made a derived unit. "We 

 can then determine from equation (31) the ratio of the units 

 in question in the old and new systems. 



If, for example, we wish to pass from the last system with 

 four fundamental units to system (17), where [l], [t~\, and 

 [k] are fundamental units, we must make [m] a derived unit 

 and eliminate the constant c 2 from the equation k = c 2 7na. We 

 accomplish this by putting c 2 m=m 1 , whence 

 [m]=n[m 1 ], 



where n is the numerical value of c 2 . The new unit of mass 



[t 2 k~\ 

 ~r • 



If we retain the last units, we have 



_ 1 sec. 2 x gramme (Paris) 



° 2 ~ 0-009808 metre x kilogramme ' 

 and 



-. , ., 1 sec." x gramme (Paris), 



1 kilogramme = - ^ nono -. — 



& 0*009808 metre 



(32) 



.. sec. x gramme (Paris) ,. ™n™n i -t 



1 e v 1 =0-009808 kilogramme. 



metre & 



If we wish to make \t~\ and not \m~\ a derived unit, we must 

 put c 2 a=a 1} so that the equation k = c 2 ma becomes Jc = ma 1 , 

 defining the conception a x . 

 We obtain then 



[«]= fl W; 



and by introducing the dimensions of [r/] and [«]], 



whence 





