Physical Units of Nature. 389 



units deserving of the title of a truly Natural Series of Phy- 

 sical Units. 



11. It now only remains to deduce the units of length, time, 

 and mass belonging to this series. For this purpose we may 

 use dimensional equations. Remembering, as is well known, 



that the dimension of a unit of velocity is L L that of a unit 



[L 3 1 

 ^r-jTT^ , and that of an electro- 

 magnetic unit of quantity [vLM], we find from equations 

 (1), (2), and (3) respectively that 



% =4 (4) 



=bA_ ? (5) 



M/r; m x t 



V / I^ 1 = Cx/7^ 1 ; (6) 



in which L 1? M 1? and T x are used to designate the units of 

 length, mass, and time in the " Natural " series ; while l 1} m 1} 

 and t x represent the corresponding units in the metric series, 

 viz. the metre, gramme, and second. A, B, and C also are 

 used, for brevity, to stand for the numerical coefficients of 

 equations (1), (2), and (3) ; viz. for the numbers 3 VIII, 



and 



3 XIIF XXII 



Solving equations (4), (5), and (6), we find 



Ll A" 1 ' ' * 



(?) 



1 £2 ■ h) (o) 



,, CA 



Ml= 71-i 00 



Substituting for A and B their numerical values, and wri- 

 ting metre, second, and gramme for l 1} t 1} ?n 1} 



Ll=C ^7i5XTv metres ' 



Tl=c B37ilxiri seconds ' 



M X =C 3^15 XIV grammes ; 



or, more simply (inasmuch as 10 is sufficiently near to 3 \/lo 

 to be used instead of it in an approximation like the present), 



