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XXVII. On the Relation of the Dimensions of Physical 

 Quantities to Directions in Space*. By W. Williams, 

 Assistant in the Physical Laboratory, Royal College of 

 Science]. 



IN a paper read before the Physical Society, Nov. 24, 1888, 

 Prof. Pucker showed that in the dimensional formulae of 

 electromagnetic qualities fi and k the two specific capacities 

 of the medium are generally omitted, or rather their dimen- 

 sions are suppressed, and that the consequence of this sup- 

 pression is to give rise to two artificial systems of dimensions, 

 the electrostatic and the electromagnetic. That these systems 

 are artificial appears when we consider that each, apparently, 

 expresses the absolute dimensions of the different quantities, 

 that is their dimensions only in terms of L, M, and T; whereas 

 we should expect that the absolute dimensions of a physical 

 quantity could be expressed only in one way. Thus, from the 

 mechanical force between two poles, we get 



and this, being a qualitative as well as a quantitative relation, 

 involves the dimensional identity of the two sides. If now 

 fi is put =1, we either ignore its dimensions or assume that 

 it has none, being but a mere number. In the former case 

 the dimensions deduced for m under such circumstances must 

 be admittedly artificial, since if the suppressed dimensions of 

 fju were inserted, those of m would be different, and would 

 involve a different physical interpretation. In the latter case 

 the dimensions deduced for m must be its absolute dimensions 

 and must therefore involve its physical interpretation. But 

 if we start with the relation 



'-$) 



q = r \/kf 



where/ is now the force between two charges q, and if we 

 treat k as a pure number, we obtain for m by means of the 

 dimensions of q deduced by this relation absolute dimensions 

 again, but different now to what we had before. In this way 

 we get two different absolute dimensions for the same physical 

 quantity, — each of which involves a different physical inter- 

 pretation. If, however, /j, and k be not mere numbers, but 

 concretes expressing physical properties of the medium, as is 



* See Note at the end of the Paper. 



t Read before the Physical Society, June 24, 1892, and communicated 

 by A. W. Riicker, F.R.S. 



