DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 313 



= F - VZm + aZmr, - V2m' - a'Lm'r,', ~\ 



= - F.(b- h) + r2ra»/-«2 \m(? + v°)},> (3). 



m - VZm'r,' - u'Zm'W* + ?'•). ) 



If M = 2m = mass of the carriage without the hind wheels and axle ; 

 M' = 2»?' = mass of the hind wheels and axle; 

 k the radius of gyration of M about its own centre of gravity ; 

 &' that of M' about its axle. 



Since 2 m r\ = Mb, 

 •2m'ri= 0, 



S»(f + >/ 2 ) - Mia 1 + ¥ + k*) t 

 2z»'(p + v*) = M'k'*, 

 the equations (3) above reduce themselves to 

 = F - {M + M') V + Mba, 



= - F(b -h) + MbV- M{a % + ¥ + A 8 ) a, 



0= - clM'k'K 



The last gives a = 0, which shows, as we might have expected, 

 that the rail, being perfectly smooth, has no power of impressing a 

 finite angular velocity by a horizontal reaction. 



The two former give by the elimination of V, 



h M ' h 



F M + W 



a = 



M „ Mb* 



a % + ¥ + 



M+M' 



If the rail be perfectly rough, the equation (2) becomes by the 

 above reductions, 



0= {F - (M + M') F + Mba] $s 



+ \Fh - Fb + MbV - Mia 9 + 6 2 + k*) a] $6 



- a'M'k'*W, 



