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CHAPTER V. 

 GENERALIZATION OF THE PRINCIPLES OF MECHANICS. 



Sect. 1. Generalization of the Second Lam of 

 Motion. Central Forces. 



rilHE Second Law of Motion being proved for 

 J_ Constant Forces which act in parallel lines, 

 and the Third Law for the Direct Action of bodies, 

 it still required great mathematical talent, and 

 some inductive power, to see clearly the laws which 

 govern the motion of any number of bodies, acted 

 upon by each other, and by any forces, anyhow 

 varying in magnitude and direction. This was 

 the task of the generalization of the laws of mo- 

 tion. 



Galileo had convinced himself that the velocity 

 of projection, and that which gravity alone would 

 produce, are " both maintained, without being 

 altered, perturbed, or impeded in their mixture." 

 It is to be observed, however, that the truth of 

 this result depends upon a particular circumstance, 

 namely, that gravity, at all points, acts in lines, 

 which, as to sense, are parallel. When we have 

 to consider cases in which this is not true, as when 

 the force tends to the center of a circle, the law 



