THE LAWS OF MOTION 309 



accelerations, should always be zero. But the only way in 

 which these accelerations can be zero is seen at once from 

 (iii.) to arise from in^ and 7«^, or the masses of A and B, being 

 equal, for then the difference di^ — nii, is zero. Thus rest 

 will depend on the equality of the masses of A and B. 



A further conceptual notion can now be introduced, 

 namely, that the terminal physical effects — consequent 

 sense-impressions — are not altered in magnitude, only in 

 direction, by carrying a weightless inextensible string 

 round any " perfectly smooth " body. This again is a 

 purely conceptual limit to a very real perceptual experi- 

 ence. Now we will suppose our string placed round a 

 perfectly smooth horizontal cylinder or peg inserted under 

 it at its mid-point C, so that the portions ^A, /B of the 

 string hang vertically downwards. We can further sup- 

 pose that the particular systems, which produce the 

 acceleration g in both A and B, are now replaced by the 

 single system of the earth, for Galilei has demonstrated 

 that all particles at the same place on the surface of the 

 earth are to be conceived as having the same vertical 

 acceleration {g) towards the surface. We conclude, 

 therefore, that if two particles be connected by a weight- 

 less inextensible string placed over a perfectly smooth 

 cylinder, the acceleration of one downwards and the other 

 upwards is given by the relation (iii.) and the tension in 

 the string by (iv.). Hence, if the particles are to be at 

 rest, or to " balance each other," their masses must be 

 equal. In this case, since ni^ = 7U^, the tension in the 

 string equals m^xg, or equals the product of the mass 

 of A into the acceleration of A due to the earth ; that is, 

 equals the force of the earth on A. This force is termed 

 the weight of A, and since in^ = m^, it follows that the 

 weight of A is equal to the weight of B. 



In this investigation, therefore, we have reached the 

 simplest conceptual notion of a weighing-machine — an 

 inextensible string, with the particles suspended from its 

 extremities, placed over a smooth cylinder. If the 

 weights of the particles are equal, their masses will also 

 be equal, and they will balance. Thus equality of masses 



