238 Mr. W. Williams on the Relation of Dimensions 



the manner the unit length enters their definition is different : 

 in the case of work the two units of length involved are in 

 the same direction ; in the case of couple they are mutually 

 at right angles. The absolute measure of a force is the work 

 done through unit linear displacement ; similarly, the absolute 

 measure of a couple is the work done through unit angular 

 displacement. Hence the relation between couple and work 

 is similar to that between force and work, the difference 

 being that angular displacements are considered in the 

 former case, linear displacements in the latter. But the 

 measure of an angular displacement is independent of the unit 

 length. Hence, in expressing the numerical dependence of 

 the unit couple and the unit of work upon the fundamental 

 units we get the same formula ML 2 T -2 , although the difference 

 in the physical nature of the quantities is of the same kind as 

 that between force and work. And generally, when used in 

 the purely numerical sense above indicated, the dimensional 

 formulse fulfil all requirements ; it is only when endowed with 

 the higher function of defining the physical identities of the 

 various quantities that they are found to fail. 



That the dimensional formulas are regarded from this higher 

 standpoint — that is, regarded as being something more than 

 mere " change ratios " between units — is shown by the fact 

 that difficulties are felt when the dimensions of two different 

 quantities, e. g., couple and work, happen to become the same. 

 If, however, the numerical dependence of the units of the two 

 quantities upon the fundamental units be the same, and if the 

 formulae are to express nothing more, then the two quantities 

 must have the same dimensions, and from this point of view 

 we are not entitled to feel any difficulty in the matter. That 

 such difficulties are felt arises therefore from the more com- 

 prehensive signification which is attached to the formulae — a 

 signification which obviously includes all the numerical consi- 

 derations which alone constitute the more restricted one. 



Let us, therefore, for the purpose of the present paper, 

 regard the dimensional formula of a quantity as the symbolical 

 expression of the physical nature of that quantity, so far, of- 

 course, as it depends upon the fundamental conceptions of 

 mass, space, and time (and, in the case of thermal and elec- 

 trical quantities, of secondary conceptions also ultimately 

 dependent upon mass, space, and time). To obtain the 

 formula for any quantity, it is only necessary to express how 

 the unit of the quantity is built up from the fundamental and 

 secondary units. Now, the units of mass and time, and all 

 secondary units, are involved in all physical units in a simple 

 manner. They are raised to different powers. But, owing 



