68-72] NOTION OF MASS. 87 



70. Mass-ratio. Let v and v r be the volumes of two material 

 figures, f and f the magnitudes of the mean accelerations pro 

 duced by v f in v and by v in v ; then the ratio f : f is defined 

 to be the mass-ratio of the two figures v and v, and/:/ is the 

 mass-ratio of the two figures v and v. We make it a rule that 

 the mass-ratio of two material figures is independent of the time. 



Suppose v&quot; is the volume of a third material figure. Suppose 

 that by the mutual action of v&quot; and v there are produced in v and 

 v&quot; mean accelerations of magnitudes / and /&quot; relative to the same 

 frame as before. 



Suppose that by the mutual action of v&quot; and v there are 

 produced in v and v&quot; mean accelerations of magnitudes /, and /,&quot; 

 relative to the same frame as before. 



Then we make it a rule that these are not independent but 



f f&quot; f 

 connected by the relation ^ . J -~- . = 1. 



/ /i /a 



This amounts to saying that 



mass ratio of v and v , , 



-: ? i 7, mass ratio ot v and v . 



mass ratio of v and v 



71. Mass. If we associate the number 1 with any particular 

 material figure A, then we can associate a definite positive number 

 m with any other material figure B, this number is the mass-ratio 

 of the two figures A and B. We call it the mass of B. According 

 to this the mass-ratio of two figures, A and B, is the ratio of the 

 mass of A to the mass of B. 



72. Conception of a body. We imagine a body to be 

 made up of material figures* each of which is a homogeneous part 

 of the body and has a definite mass, and we define the mass of 

 the body to be the sum of the masses of the material figures of 

 which it is made up. 



To cover all cases we imagine the material figures of which a 

 body is made up to occupy infinitesimal elements of volume with 

 infinitesimal masses, and we speak of one such figure as a particle 

 of the body. 



Since the mass of a portion of a body however small may be 

 taken as the unit of mass there will be no inconsistency in the 

 idea of an isolated particle of finite mass. 



* Cf. Critical Note at the end of Chapter VIII. 



