15, 16] CHANGE OF UNITS. GRAVITATION. 29 



The unit of acceleration is one centimeter -per -second per second, 



cm. 



written -- ' 2 = cm. sec.~ 2 . (It is to be noted that in a derivative 



sec. 



such as -y^j-; the numerator being a differential of no matter what 



order is of the same dimensions as s, while the denominator being 

 the square of a differential is of dimensions [T 2 ]). 

 Since momentum = mass velocity, we have 



mr -i [Mass] [Length] rML~\ 

 [Momentum] = * rrimel = ~Y~ ' 



Since force = mass acceleration, 



[~F -, _ [Mass] [Length] __ [MlTi 

 [Time 2 ] ~" L~r*J 



The unit of force is one gram -centimeter -per -second per second. 

 It is called a dyne. 



Moment of a force being force length is of dimensions 



r 2 



The dimensions of an angular magnitude, being those of the 

 ratio of two quantities of the same kind, arc and radius, are zero. 



Angular velocity being defined as -gp- is of dimensions J 



All physical equations must be homogeneous in the various 

 units, that is, the dimensions of every term must be the same. This 

 gives us a valuable check on the correctness of our equations. 



For an excellent account of the theory of dimensions the reader 

 may consult Everett, The C. G. S. System of Units. 



16. Universal Gravitation. We may now convert the 

 kinematical statement of 12 regarding the planetary motion into 

 the dynamical one, that the sun attracts the different planets with 

 forces proportional directly to the product of their masses and in- 

 versely to the square of their distances from itself. From this we 

 may pass to Newton's great generalization: Every particle of matter 

 in the universe attracts every other particle, with a force whose direction 

 is that of the line joininy the two, and whose magnitude is directly as 

 the product of their masses, and inversely as the square of their distance 

 from each other 1 ), 



the factor of proportionality y being the same for all bodies. This 

 is the law of Universal Gravitation, 



1) Thomson and Tait, Treatise on Natural Philosophy, Part II, p. 9. 



