of Physical Quantities to Directions in Spa^e. 237 



Before doing this, however, it becomes necessary to examine 

 in more detail the real nature of dimensional equations, and to 

 determine how far they are capable of giving reliable results 

 when used in the way here suggested. Primarily, a dimen- 

 sional formula expresses only numerical relations between 

 units, and for the purpose of the present paper is defective 

 from the fact that different physical quantities have the same 

 dimensional formulae. For example, couple and work, as 

 pointed out by Prof. S. P. Thompson in the course of the 

 discussion on Prof. Riicker's paper above referred to; or, 

 again, an area and the square of the same vector-length ; pres- 

 sure and tangential force per unit area, &c. If, therefore, di- 

 mensional equations are to be used at all in the sense above 

 indicated, it becomes necessary, in the first place, to be assured 

 that the process is valid, and, in the second place, that no 

 contradictory or unintelligible results arise from causes such 

 as the above. 



The dimensional formula of a physical quantity expresses 

 the numerical dependence of the unit of that quantity upon 

 the fundamental and secondary units from which it is derived, 

 and the indices of the various units in the formula are termed 

 the dimensions of the quantity with respect to those units. 

 When used in this very restricted sense, the formulae only 

 indicate numerical relations between the various units. It 

 is possible, however, to regard the matter from a wider 

 point of view, as has been emphasized by Prof. Riicker in 

 the paper above referred to. The dimensional formulae may 

 be taken as representing the physical identities of the various 

 quantities, as indicating, in fact, how our conceptions of their 

 physical nature (in terms, of course, of other and more fun- 

 damental conceptions) are formed — -just as the formula of a 

 chemical compound indicates its composition and chemical 

 identity. This is evidently a more comprehensive and funda- 

 mental view of the matter, and from this point of view the 

 primitive numerical signification of a dimensional formula 

 as merely a change ratio between units becomes quite a 

 dependent and secondary consideration. 



The question then arises, what is the test of the identity 

 of a physical quantity ? Plainly, it is the manner in which 

 the unit of that quantity is built up (ultimately) from the fun- 

 damental units L, M, and T, and not merely the manner in 

 which its magnitude changes with those units. Thus, the unit 

 couple and the unit of work both change in the same manner 

 with the unit length, but they are physical quantities of dif- 

 ferent kinds. Their numerical dependence upon L, M, and 

 T is the same, as expressed bv the formula (MLT~ 2 )L, but 



Phil. Mag. S. 5. Vol. 34. No. 208. Sept. 1892. S 



