of Physical Quantities to Directions in Space. 239 



to the dimensions of space, the unit of length is involved jn 

 different ways, according to the different relative directions in 

 which it may be taken. In all cases, however, the nnit of 

 any quantity can be completely expressed — so far as it in- 

 volves the unit length — by taking the unit length along one 

 or more of three mutually rectangular directions Ox, Oy, Qz, 

 whose absolute direction in space is of course determined by 

 the nature of the physical relation into which the quantity 

 enters. The dimensional formulas can therefore be expressed 

 in terms of M, T, and X, Y, Z, where M is the unit of 

 mass, T the unit of time, and X, Y, Z the unit of length 

 taken respectively along the directions 0#, Oy, Oz. Thus, if 

 MXT" 2 is the unit of force, MX 2 T~ 2 is the unit of work ; 

 MXY^Z" 1 ! -2 , energy per unit volume : MXZT -2 , a couple 

 in the plane XZ, &c. The above quantities are, of course, the 

 same in kind but different in direction from MYT~ 2 , MZT _J , 

 &c. This method of expressing the dimensional formulae is 

 nothing more than a modified extension of Prof. S. P. 

 Thompson's suggestion as to the use of n/ — 1 in dimensional 

 formulae, as will appear more fully later (see discussion on 

 Prof. Pucker's paper above referred to) . 



When it is necessary to determine the numerical dependence 

 of the unit of a quantity upon the fundamental and secondary 

 units, the distinction between units of length taken in different 

 directions must be suppressed, and the unit of length in what- 

 ever direction taken must be represented by a single symbol 

 L. The index of L will therefore be the sum of the indices 

 of X, Y, and Z, and the formulae thus simplified will indicate 

 the changes in the magnitude of the unit when the funda- 

 mental and secondary units are themselves changed. Thus, 

 we can immediately deduce from any formula the particular 

 and simplified form it assumes when only the numerical rela- 

 tions between units are to be considered. It will be convenient 

 to designate these simplified forms of the formulae the " change 

 ratios " of the units, and to reserve the term " dimensional 

 formulae " for the more general forms in which the identities of 

 the quantities are primarily concerned. 



The dimensions of the ordinary aynamical quantities may 

 now be expressed ; — 



1. Inertia = M. 



2. Length = X, Y, or Z. 



3. Area = XY, YZ, or ZX. 



4. Volume = XYZ. 



5. Bensitv = MX^Y^Z - 1 . 



S2 



