INTRODUCTION. XV11 



has made copies of the kilogramme. One of these is taken as standard, and is 

 called the International Prototype Kilogramme. The others were distributed in 

 the same manner as the metre standards, and are called National Prototypes. 



Comparisons of the French and British standards are given in tabular form 

 in Table 2 ; and similarly Table 3, differing slightly from the British, gives the 

 legal ratios in the United States. In the metric system the decimal subdivi- 

 sion is used, and thus we have the decimetre, the centimetre, and the millimetre as 

 subdivisions, and the dekametre, hektometre, and kilometre as multiples. The 

 centimetre is most commonly used in scientific work. 



Time. The unit of time in both the systems here referred to is the mean 

 solar second, or the 86,4ooth part of the mean solar day. The unit of time is 

 thus founded on the average time required for the earth to make one revolution 

 on its axis relatively to the sun as a fixed point of reference. 



Derived Units. Units of quantities depending on powers greater than unity 

 of the fundamental length, mass, and time units, or on combinations of different 

 powers of these units, are called "derived units." Thus, the unit of area and of 

 volume are respectively the area of a square whose side is the unit of length and 

 the volume of a cube whose edge is the unit of length. Suppose that the area of 

 a surface is expressed in terms of the foot as fundamental unit, and we wish to 

 find the area-number when the yard is taken as fundamental unit. The yard is 

 3 times as long as the foot, and therefore the area of a square whose side is a 

 yard is 3 X 3 times as great as that whose side is a foot. Thus, the surface will 

 only make one ninth as many units of area when the yard is the unit of length as 

 it will make when the foot is that unit. To transform, then, from the foot as old 

 unit to the yard as new unit, we have to multiply the old area-number by 1/9, or by 

 the ratio of the magnitude of the old to that of the new unit of area. This is the 

 same rule as that given above, but it is usually more convenient to express the 

 transformations in terms of the fundamental units directly. In the above case, 

 since on the method of measurement here adopted an area-number is the product 

 of a length-number by a length-number the ratio of two units is the square of the 

 ratio of the intrinsic values of the two units of length. Hence, if / be the ratio 

 of the magnitude of the old to that of the new unit of length, the ratio of the cor- 

 responding units of area is P. Similarly the ratio of two units of volume will be 

 T 3 , and so on for other quantities. 



Dimensional Formulae. It is convenient to adopt symbols for the ratios 

 of length units, mass units, and time units, and adhere to their use throughout ; 

 and in what follows, the small letters, /, tn, /, will be used for these ratios. These 

 letters will always represent simple numbers, but the magnitude of the number 

 will depend on the relative magnitudes of the units the ratios of which they repre- 

 sent. When the values of the numbers represented by /, #/, / are known, and the 

 powers of /, m, and / involved in any particular unit are also known, the factor for 

 transformation is. at once obtained. Thus, in the above example, the value of/ 

 was 1/3 and the power of /involved in the expression for area is / 2 ; hence, the 

 factor for transforming from square feet to square yards is 1/9. These factors 



