296] MEASUREMENT AND UNITS. 373 



numerator is a number expressing a velocity and the denominator is a 

 number expressing an interval of time ; force is measured by the product 

 of a number expressing a mass and a number expressing an acceleration ; 

 and all the other magnitudes that occur are in similar ways dependent 

 upon lengths, times, and masses. 



295. Dimensions. A number which expresses a quantity is said to be 

 of one dimension in that quantity. If the unit of measurement is altered 

 so that the new unit is a certain multiple x of the old, the number expressing 

 the quantity in terms of the new unit is the quotient by x of the number 

 expressing the quantity in terms of the old unit. 



The number expressing a derived quantity is, in every case, the product 

 of three numbers A, B, (7, of which A is a homogeneous expression of degree 

 p in numbers expressing lengths, B is a homogeneous expression of degree q 

 in numbers expressing intervals of time, and C is a homogeneous expression 

 of degree r in numbers expressing masses. We say that the quantity is of 

 p dimensions in length, q dimensions in time, and r dimensions in mass. 

 We express this shortly by saying that the dimension symbol of the quantity 

 is [L] p [ T] q [i/] r . The numbers p, q, r may be positive or negative, integral 

 or fractional, or zero. 



If the units of length, time, and mass are changed so that the new units 

 are respectively #, y, z times the old, the measure of any quantity in terms 

 of the new units is obtained from its measure in terms of the old units by 

 dividing by x p y (l z r ^ where [Z]p[7 T ][J^] 1 ' is the dimension symbol of the 

 quantity. 



The condition that a mathematical equation or inequality between numbers 

 expressing quantities may be a valid expression of a relation between the 

 quantities is that every term in it must be of the same dimensions. 



296. Physical Quantities. We give here a list showing the principal 

 derived quantities that occur in Dynamics and their dimension symbols. 



Velocity M 1 ^]- 1 . 



Acceleration [L] 1 [ T] ~ 2 . 



Moment of Momentum') , r 12 r/7Ti-iri/ii 



Impulsive Couple } L 



Kinetic Reaction^ - t m 



Force J 



Kinetic EnergyJ [I]2[7 



Power [LY[T} 



Density [L\~*[Mf. 



Constant of Gravitation [L] 3 [ T] ~ 2 [ M] 



