240 Mr. W. Williams on the Relation of Dimensions 



6. Velocity =XT~\ YT" 1 , ZT" 1 . 



7. Acceleration = XT~ 2 , YT~ 2 , ZT~ 2 . 



8. Linear momentum = MXT" 1 , MYT" 1 , MZT" 1 . 



9. Force = MXT" 2 , MYT" 2 , MZT~ 2 . 



10. Work = MX 2 T" 2 , MY 2 T~ 2 , MZ 2 T~ 2 . 



11. Energy per unit volume = MXY _1 Z _1 T~ 2 , 



MX-'YZ-'T- 2 , MX-'Y-'ZT" 2 . 



12. Pressure = MXY-'Z^T" 2 , [Force = MXT~ 2 , 



per unit of surface YZ] &c. 



Thus, the dimensions of pressure are the same as those of 

 energy per unit volume, as should be the case, for the pressure 

 at a point in a gas is given by p = i pV 2 , where p is the den- 

 sity and pV 2 is of the dimensions of energy per unit volume. 

 The case is different from that of W = G#, where W is the 

 work done by a couple G through an angular displacement 0. 

 In the former case, p is the same in kind as pV 2 , nothing 

 concrete intervening. In the latter case, G cannot be the 

 same as W unless be a mere number, having no reference to 

 anything concrete, which is not the case. 



13. Couple = MXYT~ 2 , (Force along X or Y, Arm 



Y or X) &c. 

 The rational measure of an angle is -, where s is the arc 



described by the radius r rotating about one extremity. Let 



"ds be an element of the arc, then s = Sds, and = , or 



"$6 = — . Let r be measured along X, then, since "ds is always 



an element at right angles to r, "fts will be measured along Y 

 or Z. If these directions (X, Y, Z) be carried along with r — 

 that is, if we take instantaneous axes at every point of the 

 arc — the axes bearing the same relation to the radius and 



tangent at every point, we get ^# = — , and = — ~. To 



express this dimensionally, we have to neglect the summation 

 X ; for a dimensional formula expresses the nature of a quan- 

 tity, not its magnitude, and the same formula must therefore 

 apply to both 6 and ^6. The dimensions of and ~d0 are 

 therefore YX" 1 . 



14. Angles = XY -1 , (Direction of radius Y, of the 



arc X) &c. 



15. Angular Velocity = XY^T -1 , Ac. 



