246 Mr. W. Williams on the Relation of Dimensions 



purposes, a pure number may be defined as the ratio of two 

 concretes of the same kind. If the concretes are directed 

 quantities, their ratio is not a pure number unless they are 

 similarly directed. Thus, the ratio of a force along X to that 

 along Y is of the dimensions (MXT-^MYT-^XY" 1 ; that 

 is, of the dimensions of an angle, or rotation, indicating that 

 to compare the scalar magnitude of the two forces, they have 

 to be rotated into coincidence. And, generally, the ratio of 

 any two directed quantities of the same kind is a pure number 

 together with a quantity of the nature of rotation, the latter 

 being the versor part of the quotient of two vectors. Since 

 the measure of a rotation is independent of the unit length, 

 the versor part of the quotient is a pure number so far as the 

 scalar unit length is concerned. It cannot, however, be truly a 

 pure number, for different degrees of rotation are compared 

 in terms of definite units of rotation. Hence, a versor, or its 

 equivalent, an angle, or an angular displacement is a concrete 

 quantity equally well with mass, length, and time, and should 

 have its own proper dimensions. It is only because of the 

 restriction of the term " concrete " to quantities the magni- 

 tudes of whose units change with the units of length, mass, and 

 time that concretes of the nature of angles, and angular dis- 

 placements, come to be regarded as purely numeric. From 

 this point of view, therefore, a pure number is always of the 

 nature of a tensor, or rather, unity must be regarded as the 

 dimensional formula of a tensor, and of physical quantities 

 of like nature, e. g. volume-strains *. 



Let a, /3, 7 be three vectors (non-coplanar) . Then 



1. a., /3, or 7 = Linear displacements, or lengths in magni- 

 tude and direction. 



2. V(a/3) &c. = Area of parallelogram bounded by a, /3. 



3. S (ay&y) — Volume (neglecting sign) of parallelopiped 

 defined by a, /3, 7. 



4. U^= Versor-Q= Angular displacement required to 



bring /3 parallel to a. 



If a, j&, 7 be vectors mutually at right angles, it is un- 

 necessary to distinguish between scalar and vector products, 

 •and if they be also equal, between tensors and versors, for the 

 product of any two vectors is now a vector, and of three a 

 scalar : the quotient of any two is a versor, since the tensor 

 = 1 ; while the product of parallel vectors is always scalar. 



We may therefore take a, a/3, a/37, p> anc ^ ~~ ( or un ity) as 



* See "The Multiplication and Division of Concrete Quantities/' 

 Prof. A. Lodge, 'Nature,' July 19, 1888. 



