262 Mr. W. Williams on the Relation of Dimensions 



linear displacement at a point in the field upon which depend 

 all the electromagnetic forces and flaxes at that point. In a 

 similar way, if p involves R~ 2 , ra, B, E, E would be indepen- 

 dent of R, and e, D, H, C, C dependent upon it, leading to 

 the same conclusion as before. Thus, p and k must be quan- 

 tities independent of the electromagnetic reactions (specified 

 by R) which may be going on in the field, as is otherwise 

 evident since p and k express physical properties of the 

 medium. We have therefore to consider the different ways 

 the dimensions of p (say) can be built up from M, X, Y, Z, 

 and T, subject to the conditions already laid down. 



The possible dimensional values for p are to be obtained 

 from M* 1 X ±l Y ±l Z~ l T°' +2 , by forming all the possible com- 

 binations of the quantities taken all together. The possible 

 cases are : — 



I. 1. MX-'Y-'Z- 1 . II. 1. MX-'Y-'Z-'T 2 . 



2. MXYZ" 1 . 2. MXYZ _1 T 2 . 



3. MYY-'Z" 1 . 3. MXY^Z^T 2 . 



4. MX-'YZ- 1 . 4. MX-'YZ-'T 2 . 



III. 1. M-'X-'Y^Z- 1 . IV. 1. M^X^Y^Z^T 2 . 



2. M-'XYZ" 1 . 2. M-'XYZ-'T 2 . 



3. M-'XY-'Z" 1 . 3. M-'XY-'Z-'T 2 . 



4. M^X^YZ" 1 . 4. M^X^YZ^T 2 . 



There are thus sixteen cases, and since for each value of /jl 

 we have up to the present supposed that II may ultimately 

 coincide with X, Y, or Z, or with neither, the different sys- 

 tems which may be deduced from the above are sixty-four 

 in number. There are other conditions, however, which enable 

 us to reduce this number. 



The quantities m, E, C, and e are scalar, and B, E, C, D 

 vectors. Hence, the dimensional values of p must be such 

 that when substituted in the dimensional formulae, the scalar 

 quantities remain scalars, and the vector quantities vectors 

 properly directed with respect to the dimensional axes. Since 

 all quantities are scalar so far as M and T are concerned, it 

 is only necessary to examine how the scalar and vector 

 character of the quantities depend upon X, Y, Z. To do 

 this, we may take the relation [mJ^fMRT-^X^Y^Z 1 )], 

 and substitute for p successively the four factors (XYZ) -1 , 

 (XYZ" 1 ), (XY^Z- 1 ), (X-'YZ- 1 ). The conclusions de- 



