DIMENSIONS OF A DERIVED UNIT. 585 



as a function of the three fundamental units of length, of time, 

 and of mass. 



A system of measurements based on these principles is called 

 an absolute system, the word absolute being employed in contra- 

 distinction to the term relative, which would characterize a system 

 of measurements independent of each other. 



604. DIMENSIONS OF A DERIVED UNIT. Let n be the numerical 

 expresssion of a quantity, that is the number of units which it 

 contains, and [N] the magnitude of the unit of comparison ; if this 

 number be taken equal to [N'], the magnitude to be measured 

 will be expressed by another number ri , and we shall have the 

 ratio 



which gives 



jjfJNj 



n [NT 



It follows from this that the ratio of the numerical values of a 

 given quantity is equal to the inverse ratio of -the magnitudes which 

 have served to measure it. 



When the unit is a derived unit, and it varies in consequence 

 of a change in the magnitude of the fundamental unit, in order to 

 learn this latter ratio we must know in what manner the derived 

 unit depends on the fundamental units. The relation of a derived 

 unit to the fundamental units determines the dimensions of this unit. 



We shall represent the fundamental units of length, mass, and 

 time, by the symbols [L], [M], and [T], and the magnitude of any 

 unit by a letter enclosed within a square bracket [x]. The dimensions 

 of the unit of surface will be represented by the symbol [L 2 ], and 

 those of volume by the symbol [L 3 ] ; that is to say that the 

 unit of surface varies as the square, and the unit o'f volume as 

 the cube of the unit of length. 



More generally if the dimensions of a derived unit are expressed 

 by the symbol [L*MT r ], and if we take the values L, M, T, and 

 L', M', T' successively, as fundamental units, the ratio of the derived 

 units in the two systems will be 



[N] \L \M T 



