260 Mr. W. Williams on the Relation of Dimensions 



we cannot analyse the corresponding ideas to anything 

 simpler. We should thus be unable, according to any physical 

 theory, to give interpretations to formulae involving frac- 

 tional powers of the fundamental units. Our only hope in 

 such cases would be that the units themselves might cease to 

 be fundamental. On the other hand, the building up of a 

 derived from fundamental units is always a simple process, 

 nothing but ivhole units of the latter kind being involved. 

 We should thus expeet, both from the algebraical meanings of 

 integral and fractional powers, and from the manner physical 

 units are derived, to find the dimensional formulas of physical 

 quantities free from fractional powers. That the formula) of 

 all dynamical quantities, that is all the absolute formulae we 

 arc acquainted with, are of this kind, is an argument in favour 

 of this view. 



A vector length has two properties, direction and mag- 

 nitude. When squared, a vector length becomes scalar. 

 A vector enters into physical relations either as a vector 

 (having direction and magnitude) or as a scalar (having 

 magnitude only). Hence we may say that the first and 

 second powers of a vector, X and X 2 (say), represent the 

 two different ways in which it enters all physical relations. 

 For all other integral powers differ from these only in scalar 

 magnitude. If the dimensional formulae are to express dif- 

 ferent natural relations between quantities, then, so far as the 

 unit length is concerned (the representative of space), all these 

 relations are involved in X and X 2 , X being a vector. If this be 

 true, then the fact that the indices of vector lengths, in the 

 formulae of dynamical quantities, are never higher than + 2 

 becomes explained. In a similar manner, we may explain the 

 fact that the indices of M are always + 1, for here no new idea 

 is introduced by M 2 as in the case of vectors: it differs from M 

 only in scalar magnitude, and still conveys the same physical 

 idea, namely inertia. It is widely different in the case of 

 time, for every new negative power of T introduces a new 

 physical idea with respect to time — that is, it introduces the 

 conception of a new time-flux. 



It is not necessary, however, to postulate anything as to 

 these matters. It is sufficient for our present purpose to 

 know what the limits of the indices of the fundamental units 

 are in the case of known dynamical quantities, without 

 having to account for the same. The above are only sug- 

 gestions.] 



Let the dimensions of /u, be 



M = M wl R r T'(X%Y*,Z*), 



