396 PROCEEDINGS OF THE AMERICAN ACADEMY 



XXVII. 



ON THE INFLUENCE OF INTERNAL FRICTION UPON THE 

 CORRECTION OF THE LENGTH OF THE SECONDS' PEN- 

 DULUM FOR THE FLEXIBILITY OF THE SUPPORT. 



By C. S. Peiuce. 



[Communicated by the authority of the Superintendent of the Coast Survey.] 



It has been shown by Professor A. M. Mayer that the only sensible 

 resistance to the motion of a tuning-fork is proportional to the 

 velocity. In the case of a slowly vibrating body, the chief effect is 

 probably clue to that lagging of the strain after the stress, which Weber 

 has called the elastic after-effect {Nachwirkung). The influence of the 

 former mode of resistance upon the period of oscillation of a pendu- 

 lum oscillating on an elastic tripod is easily calculated. The same 

 thing cannot, in my opinion, be effected for the other kind of resistance, 

 in the present state of our knowledge ; nevertheless, the main charac- 

 teristics of the motion may be made out. Put 



t, for the time ; 



qp, for the instantaneous angle of deflection of the pendulum ; 



s, for the instantaneous horizontal displacement of the knife-edge from 

 its position of equilibrium, in consequence of the flexure of the 

 support ; 



Z, for the length of the corresponding simple pendulum ; 



h, for the distance from the knife-edge to the centre of mass of the 

 pendulum ; 



g, for the acceleration of gravity ; 



y, for the ratio of g to the statical displacement of the point of sup- 

 port, which would be produced by a horizontal force equal to the 

 weight of the pendulum ; 



a, for the coefficient of internal friction supposed proportional to the 

 velocity. 

 Then the differential equations are 



/D-,(jp -|- D'V = — 5"P 



hJy^iCp -j- D-,s = — ys — aD,5. 



