326 THE PRINCIPLES OF SCIENCE. [CHAP. 



mass, symbolised by M ; according to the view here taken 

 we may say that the dimensions of M are L 3 D. 



Introducing time, denoted by T, it is easy to see that 



the dimensions of velocity will be or LT~ l , because 



the number of units in the velocity of a body is found 

 by dividing the units of length passed over by the units 

 of time occupied in passing. The acceleration of a body 

 is measured by the increase of velocity in relation to 

 the time, that is, we must divide the units of velocity 

 gained by the units of time occupied in gaining it ; hence 

 its dimensions will be LT~ 2 . Momentum is the product 

 of mass and velocity, so that its dimensions are MLT~\ 

 The effect of a force is measured by the acceleration 

 produced in a unit of mass in a unit of time ; hence the 

 dimensions of force are MLT~ Z . Work done is pro- 

 portional to the force acting and to the space through 

 which it acts ; so that it has the dimensions of force with 

 that of length added, giving MU-T-\ 



It should be particularly noticed that angular mag- 

 nitude has no dimensions at all, being measured by the 

 ratio of the arc to the radius (p. 305). Thus we have the 

 dimensions LL~ l or L. This agrees with the statement 

 previously made, that no arbitrary unit of angular mag- 

 nitude is needed. Similarly, all pure numbers expressing 

 ratios only, such as sines and other trigonometrical func- 

 tions, logarithms, exponents, &c,, are devoid of dimensions. 

 They are absolute numbers necessarily expressed in terms 

 of unity itself, and are quite unaffected by the selection of 

 the arbitrary physical units. Angular magnitude, however, 

 enters into other quantities, such as angular velocity, which 



has the dimensions - or T- 1 , the units of angle being 



divided by the units of time occupied. The dimensions of 

 angular acceleration are denoted by T~*. 



The quantities treated in the theories of heat and 

 electricity are numerous and complicated as regards 

 their dimensions. Thermal capacity has the dimensions 

 ML' 3 , thermal conductivity, ML~ 1 T~\ In Magnetism 

 the dimensions of the strength of pole are M*L*T~ l , 

 the dimensions of field-intensity are M ^L~^T~~ l , and the 



