of Physical Quantities to Directions in Space. 24:7 



the typical representatives of the respective ideas of vector 

 lengths, areas, volumes, angles, and pure numbers (tensors). 

 If, now, we call these the dimensional formulae (that is, the 

 physical representations) of these ideas, we see that the tensor, 

 although a gwasz'-physical quantity like the versor, becomes 

 physically represented according to the above coiYventions by 

 unity, while the versor is represented by the ratio of two 

 vectors. Thus, the meaning of the fact that tensors and phy- 

 sical quantities of like nature are of no dimensions, is that 

 their dimensional representation according to the above con- 

 ventions simplifies down to unity. No other physical quan- 

 tities are, however, dimensionally represented by unity. We 

 may, therefore, say that tensors &c. are also physical quan- 

 tities, and therefore concrete, and that since every physical 

 quantity consists of two factors, namely a pure number and a 

 concrete, we may regard a pure number when occurring alone 

 in a physical relation as a physical quantity whose concrete 

 factor is a tensor. Since, however, the term " concrete " is 

 reserved for those quantities which cannot be completely 

 specified by pure numbers, we come ultimately to regard a 

 tensor as a pure number, and the versor as a concrete 

 quantity. 



If we substitute for a, ft, y in the above X, Y, Z respec- 

 tively, we get the same expressions for lengths, areas, volumes, 

 and angles as already obtained in the dimensional formulse. 

 Hence, we see that the conventions made as to the dimen- 

 sional representation of the various quantities, namely, that 

 areas, volumes, and angles should be represented by products 

 and quotients of different vector lengths instead of by powers 

 and quotients of a given one, are justified by the fact that they 

 are consistent with the meanings of products and quotients 

 of rectangular vectors. For the products and powers of 

 parallel vectors can never represent areas or volumes, and 

 their ratios can be nothing but pure numbers. 



Dimensional formulae may be conveniently expressed in 

 this way, that is, by taking three rectangular vectors in space 

 as axes of reference. We may, of course, suppose X, Y, Z 

 to be the vectors. Then the formulae already deduced will 

 be unaltered, except that the proper signs have to be inserted 

 according to the order the products and quotients are per- 

 formed. The great advantage of thus supposing X, Y, Z to 

 be vectors instead of mere Cartesian lengths is that the dis- 

 tinction between scalar and vector quantities is made apparent 

 by the formula. Thus, X 2 , Y 2 , Z 2 , or R 2 (R being any vector) 

 are scalars; X, Y, Z, XY, YZ, ZX, XY~\ YZ~\ ZX" 1 and 

 their reciprocals are vectors ; and (XYZ) a scalar, To make 



