34 Professors Perry and Ayrton on a neglected Principle 

 from (1). If the velocity of the box is uniform, 



dt 2 U ' 



therefore the relative motion of M about the centre of the box 

 is a simple harmonic motion. 



Let the box have a uniform horizontal acceleration a, then 



therefore the body M has a simple harmonic motion about a 



point at a distance -g behind the centre of the box. 



Now, whatever be the forces acting on the box or the ball, 



d\y-z) tfy d 2 z 

 dt 2 ~ dt 2 dt 2 ' 



or the acceleration of the ball relative to the box equals the 

 absolute acceleration of the ball minus that of the box. 



Let M be resisted with a frictional force proportional to its 

 mass and to its velocity relative to the box, let 2/ be the fric- 

 tional coefficient, and let the earthquake vibration be a regular 

 harmonic motion about the fixed point ; then 



^(jLp) = -2/fcf) _„»(y _.) + „« Acos (V+B), 



where A is the amplitude of the earthquake vibration, and 



2rr 



Ti being the periodic time of the earthquake vibration. If 

 when the time is nought the box is at the limit of its swing, 

 then B is nought, or 



%£> = -2/fc> -nHy- t ) + nl A cos n x t ; 



from which, substituting x for y — z, we get 

 d 2 x ~„dx 9 2 

 W~ *db"~ n + l C ° S ?h 



as the equation of relative motion of the centre of M. Now 



4tt 2 A 

 the maximum acceleration of the box is n^A, or —^ — ; con- 



sequently, if this acceleration were constant, and if there were 

 no friction impeding the motion of M, the mean position of 

 the centre of M would be behind the centre of the box bv a 



