361 



external influence, and if all the intermediate particles be fairly 

 started with the velocities appropriate to their positions in the 

 series, the constrained vibrations of the two extreme particles, aided 

 by the elasticities of the intermediate parts, are sufiicient to main- 

 tain the vibrations of those parts. 



Neither the premises of this investigation nor the conclusion have 

 the slightest reference to the problem " to discover the velocity of 

 sound.^' In order to represent the conditions of this problem, we 

 must suppose that, the row of particles being at rest, the particle at 

 one end receives a sudden impulse, and we must seek to trace the 

 manner in which this impulse is propagated along the chain ; and it 

 is evident that there is not one point of connection between Newton's 

 theory and such premises. 



Having failed in many attempts to separate the variables which 

 enter into the analysis, the author of the paper was again led to 

 consider the question by the construction of the Manchester and 

 Liverpool railway ; for the question in hand is identical in its charac- 

 ter with this one, " to investigate the effect of a concussion on a 

 train of waggons connected hy elastic btiffers ;" but although the 

 practical importance of the subject induced him to make more 

 strenuous efforts, the difficulties of the integrations again baffled him. 

 In the month of November last, however, being again led to recon- 

 sider the problem, he was so fortunate as to discover an easy method 

 of separating the variations so as to render them integrable, and 

 thus to bring the matter within the scope of strict analysis. 



The same method is applicable to problems of a higher class. 

 Thus if we suppose a number of planets, of which the attractions 

 are proportional to the distances, although these attractions be not 

 proportional to the masses of the attracting bodies, the integrations 

 can be effected. The result of the investigation shows that such a pla- 

 netary system would have as many nuclei as planets, — one of these 

 nuclei being the centre of gravity ; each of the other nuclei would 

 describe an ellipse around the centre of gravity in its own periodic 

 time ; and thus the motion of any one planet would be the com- 

 pound of as many elliptic motions, less one, as there are planets, 

 superadded to the rectilineal motion of the centre of gravity. 



It was mentioned that this is the first instance in which the pro- 

 blem of THREE BODIES has been resolved when the resultants of the 

 attractions do not all pass through one point. 



