under which a Perpettial Motion is possible. 371 



Paper, which is intended to investigate the mathematical conse- 

 quences of an assumed law, I shall not discuss the identity of 

 these supijositions : I shall only remark that the general expla- 

 nation ajjpears to be correct, and that it clears up several points 

 which always appeared to be in great obscurity. 



Let us now consider the case of a vibrating body acted on 

 by two forces, of which one is proportional to its actual distance 

 from the point of rest, and the other proportional to its distance 

 at some previous time. Putting (p (t) for the body's distance, the 

 equation is 



^^= -e.i>it)-g..pit-c). 



This equation I am unable to solve rigorously: but on the sup- 

 position that g- is small, an approximate solution may be obtained 

 from the formulae in the Memoir on the Disturbances of Pendu- 

 lums, &c. (Cam. Trans. Vol. III. p. 109.) Neglecting at first the 

 small term, we have 



whence 

 Consequently 



(p{t) = a . sin (t ,y/e + b). 



(p{t — c) = a . sin {t ^e ^- b - c ^/e), 

 and therefore /in the formulae alluded to is 

 = ag-. sin {t ^ e + b - c ,J^). 



The increase therefore of the arc of semi-vibration is 



— —j=f,ag.^\x\ {t sf^ -^ 6 - ^^n/^) cos {t sJ e + 6) 



= - ^^^j^jt {sin (2 t Jl + 2 ft - c Jl) - sin c Jli\ . 

 Vol. III. Part II. 3 B 



