m -w = - Pa ' 



SI 6 Mr POWER, ON THE PREVENTION OF THE 



vertical plane, arising from a horizontal collision, it is necessary and 

 sufficient that the centre of the buffers should be placed as nearly 

 as may be on the same level with the centre of gravity of the 

 engine or carriage. 



If we suppose the rotatory motion impressed to be very small, it 

 is easy to calculate approximately the ascent of the elevated wheel. 



M' 



Neglecting -^ and calling 9 the small angle of elevation at the 



time t, we shall have approximately, 



. d'.iaO) P 



Li L — jo" 4- 



df " *- 3f* 

 P denoting the vertical pressure at the rail, 



__ d*9 ag 



Hence -77, = — , ,* , 



dt 2 a 2 + k' 



d9 agt 



-j- = Const. j-^-yj 



dt a 2 + k 



Fh - Magt 



M{a % + k 2 ) ' 



the initial angular velocity being „, 2 7^ . 



Integrating again, so that 9 is when t = 0, we obtain 



2Fht- Magt" 



" 2 M(a 2 + k 2 ) ' 



Hence, 9 attains its maximum value when Fh — Magt = 0, or 

 t = ^rr — . and at that moment the value of 9 is 



Mag' — 2M 2 ag(a' + k 2 )' 



If we suppose the centre of gravity to be situated half way be- 

 tween the two axles, multiplying by 2 a, we find for the extreme 

 elevation of the wheels, when the disturbance is small, the expression 



F*h* 



M 2 g(a 2 +k 2 ) ; 



