of Physical Quantities to Directions in Space. 249 



MXT- 2 



Z ,„+ 



X^ 'X 



Tangential force per unit area = MY T = — ^r 



MZT -2 

 = Force along X in plane XY = vy — 



= Force along Z in plane YZ. 



These are the two components of a shearing- 

 stress referred to a unit cube. Multiplying by l %~ ' 



XZ MXZT" 2 n . ., , 



Y7 we get = Couple per unit volume. 



Now, a shearing-stress must be of the nature 



of a couple, for a shear is of the nature of an 



angle, and the product of the stress into the shear is work 



done per unit volume. 



Q , . MXT" 2 MX 2 T~ 2 _, 



burtace tension = — ^ — = — vy — = Energy per unit area. 



MXT -2 MX 2 T -2 

 Pressure = — vy = — YV7 = Energy per unit volume. 



Since, as above shown, we obtain the same dimensional 

 formulas by Cartesian and vectorial methods, we may attach to 

 X, Y, Z and their products and quotients in these formulae 

 either purely vectorial or purely Cartesian meanings just as 

 we please : in both cases, the formulas represent the same 

 physical identities. When X, Y, Z are vectors the formulas 

 express the directional properties of the corresponding quan- 

 tities. [An inspection of a formula shows this, for the pro- 

 ducts and quotients of two rectangular vectors are vectors 

 directed normally to the planes containing the original vec- 

 tors, and the products and quotients of parallel vectors are 

 scalar]. Thus: Pressure MX(YZ) _1 T _2 , scalar, for YZ is 



directed parallel to X, hence ™ is scalar. Couple MXZT -2 , 



a vector directed along Y, &c. 



In deducing the dimensions of electromagnetic quantities 

 it will be necessary to start with the dimensions of energy, 

 or energy per unit volume. Using Cartesian units of length 

 this is MX 2 T~ 2 , MY 2 T" 2 , MZ 2 T" 2 , or MR 2 T" 2 , as the case 

 may be ; and since we shall have to deal with connected 

 equations between the various quantities, the particular form 

 to be used becomes of importance. Of course, energy being 

 a scalar quantity, the above do not differ essentially : the 

 difference arises only from the different ways (the different 

 dynamical reactions) by which the expressions are derived. 



