DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 311 



same formulae for the motion of rigid bodies, as were used in the 

 preceding example. On this account I prefer treating it by the method 

 of Lagrange. 



Let us suppose the whole mass of the engine to be projected upon 

 a vertical plane parallel to the direction of the rails, and that a hori- 

 zontal impulse F from behind causes it to lift up its front wheels 

 and rotate about its hind or driving pair of wheels. 



Let x, y be the horizontal and vertical co-ordinates of any point 

 m of the carriage, deprived of its hind pair of wheels and their con- 

 necting axle, referred to a fixed origin behind the carriage. 



x\ y those of any point m of the hind wheels and axle ; 

 u, v the horizontal and vertical velocities communicated to m by 

 the impact; 



u', v' the same for m'. 



By D'Alembert's Principle, the momenta subject to the conditions 

 of equilibrium are 



F, — mu, — &c. — m'u', — &c. horizontal; 



and — mv, — &c. — m'v', — &c. vertical. 



Hence, naming x the horizontal co-ordinate of the point of appli- 

 cation of F, we have by the principle of virtual velocities, 



Fix - Z(muSx) - 2(m'u'§x') ) 



(1), 



I'v'ly')) ' 



Let V be the linear velocity communicated to the axis of the hind 

 wheels. 



a the angular velocity communicated to the carriage about the axis 

 of the hind wheels, tending to diminish x and increase y. 



a the angular velocity communicated to the hind wheels about their 

 axis, tending to increase x and diminish y. 



Id, l& any small virtual angles of rotation of the carriage and hind 

 wheels in direction of the angular velocities a, a respectively. 



8s the corresponding horizontal space described by the axle of the 

 hind wheels. 



L L2 



