,bct. ii.] DISSERTATION SECOND. 6l 



adding the products together, was the same after collision 

 that it was before it, and it was concluded with some precipita- 

 tion, by those who espoused the Leibnitian theory, that a simi- 

 lar result always took place in the real phenomena of nature. 

 Other instances were cited ; and it was observed, that a par- 

 ticular view of this principle which presented itself to Huy- 

 gens, had enabled him to find the centre of oscillation of a 

 compound pendulum, at a time when the state of mechanical 

 science was scarcely prepared for so difficult an investigation. 

 The proposition, howevcr,1s true only when all the changes 

 are gradual, and rigorously subjected to the law of continui- 

 ty. Thus, in the collision of bodies imperfectly elastic (a 

 case which continually occurs in nature), the force which, 

 during the recoil, accelerates the separation of the bodies, 

 does not restore to them the whole velocity they had lost, 

 and the vis viva, after the collision, is always less than it was 

 before it. The cases in which the whole amount of the vis 

 viva is rigorously preserved, may always be brought under 

 the thirty-ninth proposition of the first book of the Principia, 

 where the principle of this theory is placed on its true 

 foundation. 



So far as General Principles are concerned, the preceding 

 are the chief mechanical improvements which belong to the 

 period so honourably distinguished by the names of Newton 

 and Leibnitz. The application of these principles to the 

 solution of particular problems would afford materials for 

 more ample discussion than suits the nature of a historical 

 outline. Such problems as that of finding the centre of 

 oscillation, — the nature of the catenarian curve, — the deter- 

 mination of the line of swiftest descent, — the retardation pro- 

 duced to motion in a medium that resists accordins; to the 

 square of the velocity, or indeed according to any function 

 of it, — the determination of the elastic curve, or that into 

 which an elastic spring forms itself when a force is applied 



