270 On the Dimensions of Physical Quantities. 



or lower (fractional) powers of the fundamental units than 

 those encountered in the case of ordinary dynamical quantities. 



(2) All values which while satisfying (1) rendered scalar 

 magnitudes vectors, or vectors scalar. 



(3) All values which while satisfying (2) rendered the 

 direction of the instantaneous linear displacement at a point 

 in an isotropic medium parallel to Z, and therefore- normal to 

 the plane of displacement (XY). 



In the foregoiug discussion I have attempted to generalize 

 the ordinary dimensional expressions for physical quantities^, 

 and to carry somewhat further than he did the suggestion 

 of Prof. S. P. Thompson that the idea of direction may be 

 associated with the symbols as well as that of numerical 

 magnitude. This is clearly possible — as the examples given 

 show — in the case of ordinary dynamical quantities, and the 

 method certainly enables us to distinguish between things 

 which are physically different,, though (if the ordinary system 

 in which the idea of direction is suppressed be used) the 

 dimensions are the same. The dimensional formula thus 

 becomes for a physical quantity the analogue of a structural 

 formula for a chemical compound. 



Applying these ideas to the more difficult case of electrical 

 and magnetic quantities, with the actual nature of which we 

 are imperfectly acquainted, I have tried to use the method as 

 an instrument for discriminating between probable and im- 

 probable hypotheses. It is at all events, interesting to note 

 that we are thus led to tivo possible dimensional systems which 

 agree with the two principal groups of analogies by which 

 electrical and magnetic facts have been illustrated by those 

 who have most deeply studied these from the dynamical point 

 of view. Between these two I do not attempt to decide ; but 

 I cannot but hope that the method's I have suggested may 

 make the study of dimensional formulae not merely a con- 

 venient method of expressing numerical relations between 

 fundamental and derived units, but a means of seeing more 

 deeply into the physical facts they represent. 



Note. — I regret that when I communicated the above 

 paper to the Physical Society I was unaware of an important 

 article by Prof. A. Lodge on "The Multiplication and Divi- 

 sion of Concrete Quantities" (' Nature/ July 19, 1888). 

 In this article Prof. Lodge discusses the general question of 

 the products and quotients of concrete quantities, and of the 

 meaning of physical relations between concretes of different 

 kinds. He clearly points out that the dimensions of quan- 

 tities do not always afford a test of their identity, and that in 



