Systems of Physical Units. 93 



and the dimensions become 



[*,]=[•£], W-Bfj, M = [f], m-kj. 



as before. 



5. The rules we have given make it possible to pass from 

 one system of units to another. The process, however, is 

 somewhat troublesome. "When the magnitude of a physical 

 conception is given in one system of units and we wish to find 

 its numerical value in another system, the following is in 

 all cases the safest and quickest method of arriving at the 

 result * : — 



If we assume three fundamental units and denote them by 

 P, Q, E, then the dimensions of a derived unit N are of the 

 form 



X = P a Q 6 R c . 



where a, b, c are positive or negative rational exponents. If 

 we assume units of other magnitudes p, q, r, and if 7 = ep, 

 Q=fg, Jl=gr } where e.f, and g are constants, then 



If/', q. and r are fundamental units, the new derived unit n 

 has the dimensions p a , q h , r e , and 



S=e«f b g c .n, 

 or the given unit X is e a f b g c times as large as the new unit n. 

 If one of the new units, for example p, be itself a unit derived 

 from the units q. r. and a new fundamental unit s, by means 

 of the formula 



p a =q**T***, 

 then 



S = e a / b o c . p° q br r c — e a f b g c . q b+b i i*****. 

 The new unit n is determined by the formula 



n = q b + fj ir c + e i$ d ; 

 consequently in this case also 



N=c a /y,/i. 



We obtain thus the following practical rule for obtaining 

 the value of a unit referred to new fundamental units: — In the 

 dimensions of the unit replace the old fundamental units by 

 their values in the new fundamental units, and carry out the 

 algebraic operations indicated as if the names of the units 

 were algebraic magnitudes. This rule gives not only the 

 ratio of the old unit X to the new unit n, but also the dimen- 

 * Compare KolilrauicL, Leitfaden, p. 207. 



