96 THEORY OF NEWTONIAN FORCES. [PT. I. CH. I. 



Accordingly we see that the force and acceleration vanish together. 

 Integrating the equations (i), 



x di y c? 2 z c? 3 



GI C 2 C 3 



the path is a straight line, and since 



it is traversed with constant velocity. We may on the other hand 

 interpret the statement as giving us a means of measuring time. 

 Intervals of time are proportional to the corresponding distances 

 traversed by a point not acted on by forces. 



The second law gives the measure of a force. 



LEX II. Mutationem motus proportionalem esse m motrid 

 impressae, et fieri secundum lineam rectam qua vis ilia im- 

 primitur. 



Change of motion is proportional to force applied, and takes 

 place in the direction of the straight line in which the force 

 acts. 



By change of motion is meant acceleration. If we have to 

 do with different bodies, however, the factor of proportionality 

 will be different for each. 



LEX III. Actioni contrariam semper et aequalem esse reac- 

 tionem: sive corporum duorum actiones in se mutuo semper esse 

 aequales et in partes contrarias dirigi. 



To every action there is always an equal and contrary reaction : 

 or, the mutual actions of any two bodies are always equal and 

 oppositely directed. 



If we have an action between two bodies 1 and 2, if the forces 

 were proportional only to the accelerations, we should have 



ggi _ _ cgg_ a dfyi _ _ dfyz d 2 ^ _ cfrg a 

 W~ ~W ~di?~ ~W dt z ~ dt*' 



This is not the case, but we must introduce factors of proportionality, 

 so that 



