98 THEORY OF NEWTONIAN FORCES. [PT. 1. CH. I. 



The unit of mass will be assumed to be the gram, defined as 

 the one-thousandth part of a piece of platinum-iridium, deposited 

 at the place above mentioned and known as the "Kilogramme 

 Prototype" 



As the unit of time we shall take the mean solar second, 

 obtained from astronomical observations on the rotation of the 

 earth. The unit of time cannot be preserved and compared as in 

 the case of the units of length and mass, but is fortunately 

 preserved for us by nature, in the nearly constant rotation of the 

 earth. As the earth is gradually rotating more slowly, however, 

 this unit is not absolutely constant, and it has been proposed to 

 take for the unit of time the period of vibration of a molecule of 

 the substance giving off light of the standard wave-length. To 

 obtain such a unit would involve a measurement of the velocity 

 of light, which cannot at present be made with sufficient accuracy 

 to warrant the change. 



53. Derived Units. Dimensions. It can be shown that 

 the measurements of all physical quantities with which we are 

 acquainted may be made in terms of three independent units. 

 These are known as fundamental units, and are most conveniently 

 taken as those of length, mass, and time. Other units, which 

 depend on these, are known as derived units. If the same quantity 

 is expressed in terms of two different units of the same kind, the 

 numerics are inversely proportional to the size of the units. Thus 

 six feet is otherwise expressed as two yards, the numerics 6 and 2 

 being in the ratio 3, that of a yard to a foot. If we change the 

 magnitude of one of the fundamental units in any ratio r, the 

 numeric of a quantity expressed in derived units will vary pro- 

 portionately to a certain power of r, r~ n ; the derived unit is then 

 said to be of dimensions* n in the fundamental unit in question. 

 For instance, if we change the fundamental unit of length from the 

 foot to the yard, r = 3, an area of 27 sq. ft. becomes expressed as 

 3 sq. yds., the numeric has changed in the ratio 3 : 27 = 1 : 3 2 = r~ 2 , 

 and the unit of area is of dimensions 2 in the unit of length. We 

 may express this by writing 



[Area] = [Z 2 ]. 



* The idea of dimensions of units originated with Fourier: vid. Thforie 

 analytique de la Chaleur, Section ix. 



