13, 14, 15J UNITS AND DIMENSIONS. 27 



the above meter and the wave-length in air of a red cadmium ray 

 as I^S^IGSA 1 ) 



The unit of mass will be assumed to be the gram, denned 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 the accuracy with which the mean solar second 

 is known, 



15. Derived Units and 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 proportionately 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 3 sq. yds., the numeric has changed in the 

 ratio 3:27 = l: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] = [*]. 



The derived unit increases in the same ratio that the numeric of the 

 quantity decreases. In our system the unit of area is the square 



1) Travaux et Memoires du Bureau International des Poids et Mesures. 

 Tome 11, p. 85. 



2) The idea of dimensions of units originated with Fourier: Theorie ana- 

 lytique de la Chaleur, Section IX. 



