312 



Mr POWER, ON THE PREVENTION OF THE 



Then if the rail be perfectly smooth the variations 89, 89 1 , 8 s are 

 independent of each other. But if the rail be perfectly rough, so 

 that the wheel remains in perfect rolling contact with it during the 

 shock, 89" and 8s are connected by the equation 8s = r89', r being 

 the radius of the wheel. 



Again, assuming the moveable projection of the axis as a new 

 origin, let 



f , tj be the co-ordinates of m ; 



£', »/ be the co-ordinates of ml. 

 We have, manifestly, 



U mm V — r\a, 

 V = £ct, 



tf . V + n'a, 

 V' = - fa'. 



Also, 8x = 8s — ij89, 



h - m 



M = 8s + v '89', 



8y' = - f 56T. 



Again, if a, b be the horizontal and vertical co-ordinates of the 

 centre of gravity of the carriage exclusive of the hind wheels, and h 

 the vertical distance below this centre at which F passes, we have 



8x = 8s - (b - h) 89. 



Substituting these values, the equation (1) becomes 



= F {8s - {b - h) 89] - 2 {m {V - n*) (^ - r,89 )} 



- 2 {m? a89} 



- 2{m'(r+ r{d) {Is + r,'89')\ 



- 2 {m'?*a'89'\ 



Hence if the rail be perfectly smooth, equating to zero the co- 

 efficients of the arbitrary variations 8s, 89, 8&, we obtain 



(2). 



