Z&8 Mr. W. Williams on the Relation of Dimensions 



expresses only the resultant reaction taking place, namely 

 that there is a space rate of variation of the energy of the 

 system along Y, the direction of the force. The same re- 

 marks apply to the cases below » 



4. The force between two charges q is given by 



F q = m =l(£X 



MR 2 T -2 

 or diinensionally [FJ = ^ — , the space rate of variation 



of the energy of the system they constitute along X, the 

 direction of the force. 



5. The force per unit volume between two elements of 

 conductors carrying currents is given by F C = BC, where C 

 is the current-density in the one, and B the induction at this 

 one due to the other. Expressed dimensionally, this becomes* 



MR 2 T~ 2 / MRT~ 2 \ 1 

 L cJ ~ XYZ 2 "V XYZ Jz' 



which is of the dimensions of space rate of variation of energy 

 per unit volume along Z, the direction of the force, far the 

 conductors move at right angles to their lines of force which 

 are circles- about them. 



Whatever be the dimensions of p. those of k must be given 

 by jw,- 1 [Z _2 T 2 ]. Now, the ratio of the dimensions of the same- 

 quantity expressed in the two systems is a power of 

 u^fZ^T -2 ]. Hence, all possible dimensional values of p 

 together with the corresponding ones for k will, when sub- 

 stituted in the formulae, bring the two systems- into accord. 

 Of these, however, there must be one pair, and only one pair, 

 which give rise to dimensions whose interpretations are 

 physically real. 



It is, of course, impossible to determine from purely dimen- 

 sional considerations what this particular pair must be, for 

 this would imply a knowledge of the dynamical nature of 

 electromagnetism. It is possible, however, to assign to p. and 

 k dimensions fulfilling certain assumed conditions. The 

 dimensions of the remaining quantities then become unique, 

 and we may, by deducing the physical interpretations of the 

 formulas, pass on to the quantities they then represent. In 

 this way, the formulas in terms of p. and k may be utilized as 

 means for tracing out in detail the various analogies between 

 electromagnetism and dynamics. For every dimensional 

 value of ju, and k we thus obtain a perfectly connected dy- 

 namical analogue of electromagnetism, winch may or may 



