248 Mr. W. Williams on the Relation of Dimensions 



the formulae exhibit at the same time the numerical depen- 

 dence of the derived upon the fundamental units, we may 

 put X = e'L, Y=jL, Z = kL, i,j, k being quadrantal versors 

 and L the scalar unit length. Thus, neglecting signs, we 

 have : — 



Force = MXT~ 2 = M(7L)T- 2 . 



Work = MX 2 T~ 2 = M (iL) 2 T- 2 = M L 2 T" 2 . 



Couple = MXYT" 2 = MijL 2 T~ 2 = M&L 2 T~ 2 . 



Energy per unit volume, = (ML 2 T" )L~ =ML~ T~ , 



for volume = ( ijklf) = L 3 , ijk = 1 . 



MXT" 2 M/LT- 2 MT _i T _ 2/ . ..v 

 Pressure = yz = - .^ =ML T , (i=jk). 



Tangential force per unit area- MY - ■*■"" =M(/L)~ J-~ . 



Angle=XY- 1 =(*L)0'L)- 1 = 4 = | =L 



Work done by a couple = MXYT" 2 (XY _1 ) =(M2> , L 2 T" 2 )^ 

 = ML 2 T~ 2 , (y*=l). 



Thus, the indices of M, L, and T indicate the numerical 

 dependence of the derived upon the fundamental units, the 

 formulae being identical in this respect with the ordinary ones, 

 while the directional properties of the quantities are clearly 

 differentiated by **, j, k. 



Multiplying the numerator and denominator of a formula 

 by X, Y, or Z makes no change in its physical interpretation. 

 For physical quantities differ from each other only in their 

 numerical dependence upon the fundamental units, and their 

 " space relations,* 5 the former being expressed by the scalar 

 part of the formula L, M, T, the latter by the vector part i,j, k. 

 But if X, Y, Z be equal rectangular vectors, the tensors of 



y V 2 



_ — — are unity, and their versors are zero. Hence, no 



A I Z 



change can be made in the scalar or vector part of any 

 formula into which they may be multiplied. Thus : — 



^ TVrr 9 MX 2 T~ 2 

 Force = MXT~ 2 = 



X 



= Space rate of variation of energy along X. 



Force = MXT~ 2 =~ — ~ = Couple per unit area. 



