of Physical Quantities to Directions in Space. 245 



axis Z, radius X. The normal pressure due to the curvature 

 is K I - V or dimensional! y 



MYZ -l T -2 (XY -2 )==MX(YZ) ~l T -2 . 



as should be the case. 



The above examples are sufficient to illustrate the method 

 of expressing dimensional formulae in terms of X, Y, Z. In 

 all the above cases isolated quantities only are dealt with, and 

 therefore the relation of the directions of directed quantities 

 to the dimensional axes is a matter of indifference. This is 

 not so in the case of equations between quantities. Here, 

 however, we have only to transform the equations to Carte- 

 sian co-ordinates and then express the dimensions of each 

 term by means of the above. Thus, let A, B, C, . . . be 

 quantities connected by the equation 



A + B + C + D+ ... =0. 



Expressed in Cartesian co-ordinates this becomes 



(A I +A y +AJ+(B I +B^+B,) + (C.+C y + C B )+ . . . =0; 



and we can now immediately express the dimensions of every 

 term. We thus get the dimensional expression of the equation 

 itself. A physical equation implies that the quantities con- 

 nected are the same in hind. There must, therefore, be a 

 dimensional identity between the various terms. This holds, 

 however, in its wider or directive sense only of component 

 quantities in the same direction, for if we equate any term 

 in the above to all the rest we always get a term such as A^ 

 equated to another such as A , the same in kind, but different 

 in direction. If, however, we write the equation 



(A.+B.+0.+ . . .) + (A y + B y + . . .) + (A,+B,+ . . .)=0, 



there must now be a dimensional identity in the directional 

 (X, Y, Z) sense between the terms in the same bracket. 



Up to the present the dimensional formulae have been re- 

 garded ns purely conventional. It is now necessary to ex- 

 amine whether the conventions made use of can be justified 

 in any other way ; and it becomes important to have a clear 

 physical distinction between pure numbers and concretes, for, 

 according to the above conventions, dimensional formulae are 

 now extended to include quantities hitherto regarded as pure 

 numbers. 



Every concrete quantity should have definite physical di- 

 mensions in the extended sense of the term ; only pure 

 numbers should be quantities of no dimensions. For physical 



