of Physical Quantities to Directions in Space. 261 



then those of k are 



Since the indices of M, X, Y, Z in the dimensional formulae arc 

 odd multiples of J, their indices in the dimensions of jjl must 

 be odd, otherwise, on substituting for fi and k in these formulae 

 their dimensional values, the formulae would not be rational- 

 ized. For a similar reason, the indices of R and T must 

 be even. Hence, the indices of M, X, Y, Z must be ±1, 

 and of R, 0, or + 2. In the case of Z, + 1 is not admis- 

 sible, otherwise we should have Z -3 in the case of k. Thus, 

 Z must enter both //. and h as Z _l . In the case of T the 

 possible values are 0, +2, +4, . . . . ±2n. These values 

 may be tabulated thus : — 



w [*] 



(a) 



W 



l<0 



(&') 









+ 2 





+ 2 



-2 







+ 4 



+ 4 



-4 



-2 



+ 6 



+ 6 



-6 



-4 



+ 8 



+ 2/2 -2n _2(n-l) + 2(n + l) 



Under a are given the positive possible values for t in //., under 

 a the corresponding values in the case of k. Under b, the 

 negative values in the case of jjl, and under V those correspond 

 ing in the case of h. The only cases necessary to be considered 

 are T° and T 2 , for, as seen from the table, all other possible 

 values of T lead to dimensional formulae involving powers of 

 T higher than +4. 



The dimensions of /j, and k involve only X, Y, Z, M and T. 

 This is of course obvious if R has to coincide ultimately with 

 X, Y, or Z. If II is not ultimately to coincide with X, Y, or 

 Z, it cannot enter into the formulae of fi and h. For It can 

 enter into the formula of jul only as R 2 or R _ . In the former 

 case, //,* contains R, and jjT* contains R -1 . But fj? is a 

 factor of the formulae of m, B, E, E, and /a -5 of e, D, C, C, H ; 

 and since R is a factor of the formulae of all the above quan- 

 tities, we should have : — 



i. m, B, E, E, containing R in their dimensional formulae, 

 ii. e, D, H, C, C, containing R° : 



thus indicating that some of the forces and fluxes of the field 

 would be dependent upon R, the others independent of it. 

 But this is obviously impossible, since It is the instantaneous 



