264 Mr. W. Williams on the Relation of Dimensions 



that It may lie along either Y or X, and the possible systems 

 become reduced to eight, as below : — 



1. M(XYZ)" 1 . 2. M^XYZ^T 2 . 



3. MtXYZ- 1 ). 4. M-'X-'Y-'Z-'T 2 . 



5. M(XYZ)" 1 T 2 . 6. M^XYZ -1 . 



7. MfXYZ^T 2 . 8. M^X^Y-'Z- 1 . 



If the dimensions of //, be (1) those of k are (2), and if those 

 of k are (1), the dimensions of /x are (2), and similarly for 

 the other three pairs. 



The dimensional formulae 5 and 7 are unintelligible, for we 

 have no dynamical units involving positive powers of both 

 M and T, and it is difficult to give to such formula? an un- 

 telligible interpretation. The same difficulty appears in 6 

 and 8, for if one of these be the dimensional formula for fi, 

 the corresponding one for k is 5 or 7. Apart, however, from 

 this difficulty, these formulas 5, 6, 7, and 8 lead to fluid 

 theories of electromagnetism. Thus, from 5, we get for the 

 dimensions of m: — 



[m] =^[M"RT- 1 (X^Y- | Z 1 )] = M^(X- | Y- | Z- | )R(X i Y-^) 



M i T -§ T §_]y[RY -1 = M (since R here must coincide with Y). 



Similarly, from 7 we have [???] = MX 2 . Again, from 6 we get 

 [e] = M, and from 8, [e] = MY 2 . Thus, we have to suppose 

 m to be a quantity of the nature of mass or moment of inertia, 

 and similarly for e. In both cases, m and e would be pro- 

 perties of the medium depending upon its inertia, instead of 

 being parts of the electromagnetic reactions going on in the 

 medium, — an electrical current would thus be a quantity of 

 the nature of momentum, which is inconsistent with Maxwell's 

 theory of electromagnetism. Other and not less serious 

 difficulties will be met with if an attempt is made to develop 

 their interpretations more fully. 



Similar remarks apply to the systems arising from 3 and 4. 

 Thus, from 3 we have 



[^]=/^- | [M l R(X- | Y l Z- 1 )] 



= M^X-^zWRX-^Z -1 = RX" 1 = 1, 



since R here coincides with X. Thus, e is of the di- 

 mensions of a volume-strain, the ratio of two identical 

 concretes. Similarly, we get from 4, [w] = l, also a volume- 

 strain. Again, in the former case we have D = <?(YZ)~ 

 = volume-strain per unit area. The two currents C and C 



