Systems of Physical Units. 89 



other hand, the last-mentioned group may be employed for 

 this purpose in the system corresponding to equation (8). 



4. It is very important to be able to pass from one system 

 of units to another, and to determine the ratio between the 

 units of the two systems. This problem presents the following 

 special cases: — 



I. When the new system contains one new fundamental 

 unit besides the fundamental units of the old system. Let L 

 be the value in the old system of the new fundamental unit 

 defined bv the equation 



L=M*N* ; (23) 



where M and N are quantities defined independently of L. 

 As the unit for L in the new system may be chosen at plea- 

 sure, this quantity is expressed by a new number 1^; but the 

 numerical values of M and N are not altered. Equation (23) 

 is therefore not satisfied by L, M, and N, but must be written 

 with a constant a, 



L 1=a M¥ (24) 



The system of fundamental equations is therefore changed : 

 the number of equations remains the same ; but the number of 

 quantities is increased by one. Since we must regard « as a 

 new conception defined by equation (24), we must therefore 

 increase the number of fundamental units by one. Conse- 

 quently the relations of the units as well as their dimensions 

 are partially altered. By comparing equations (23) and (24) 

 we find that 



L=^, (25) 



whence, since the units are inversely proportional to the nume- 

 rical values of a given quantity, it follows that 



[L]=«[L 1 ], (25a) 



where n is the numerical value of «. 



The ratios of the remaining units are easily obtained from 

 the other fundamental equations without alteration. It is to 

 be remarked that the constant « is in general not an abstract 

 number, but that the dimensions of its unit are given by the 

 equation 



|X,]=|>M*N*3 (26) 



If, for example, we wish to pass from the system (15) into the 

 system (21), where, besides the three fundamental units [/] , 

 [t] , and [m] , we take [&] also as a fundamental unit, we must 

 start from the equation (12), which defines the conception h 



