30 I KINEMATICS OF A POINT. LAWS OF MOTION. 



The numerical value of 7, the Newtonian constant of gravitation, 

 depends upon the system of units used. Its dimensions are those of 



[Force] [Length 2 ] _ r L s "1 

 [Mass 2 ] ~\_WT*]' 



It is possible, and in astronomy is convenient to choose the 

 units in such a manner as to make y equal to unity. If this were 

 done, we should get a relation between the dimensions of mass, 

 length and time, for by supposing that y has no dimensions, we 

 should have 



Thus we should need only two fundamental units instead of three. 

 This is an example of the somewhat arbitrary nature of the dimen- 

 sions of physical quantities. What is not arbitrary however is the 

 statement that every physical equation must be dimensionally 

 homogeneous. For the purposes of physics it is customary to retain 

 the three fundamental units, giving y the dimensions specified above. 

 Determinations undertaken to ascertain the numerical value of y by 

 terrestrial observations have been made in great numbers from the 

 time of Cavendish to the present. One of the most accurate, that 

 of Boys 1 ), gives in the units which we have adopted, 



' cm - 



gm. sec. 2 



that is, two spherical masses each of mass one gram with centers 

 one centimeter apart attract each other with the force of y dynes. 2 ) 

 If two particles have coordinates x lf y lf lf x 2 , / 2 > #2 an ^ distance 

 apart r 12 , the direction cosines of the line drawn from 1 to 2 are 



and, since the force exerted by 2 on 1 has the direction of this 

 line, the equations of motion for 1 are 



A at" T! 



m* -V77T = V - 



1) Boys, Phil. Trans. 1895, I. 



2) It will be shown later that homogeneous spheres attract each other as 

 if their masses were all concentrated at their centers. 



