174 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



3. Prove that each of the lines of quickest descent in Example 2 bisects 

 the angle between the normal to the curve and the vertical at the point where 

 it meets the curve, and hence show that the line of quickest descent from 

 one given curve to another in the same vertical plane bisects the angle between 

 the normal and the vertical at both ends. 



4. When the tube (or the plane) of Article 184 is rough, and the co- 

 efficient of friction is /u, prove that the acceleration down the line of greatest 

 slope is g (sin a + p, cos a), where the upper or lower sign is to be taken accord- 

 ing as the particle is moving up or down. Also find the acceleration with 

 which a particle moves on a rough horizontal plane. 



5. Example 4 is generally assumed to apply to the motion of a railway 

 train. Consider the motion on straight horizontal rails, and let m be the 

 mass of the train, P the resultant force acting upon it apart from the resist- 

 ance, pmg the resistance, then / the acceleration with which the train moves is 

 given by the equation 



and, if P=^mg^ the train moves with uniform velocity. The force P is 

 known as the " pull of the engine," but it is not the tension in a coupling 

 between the engine and the rest of the train. 



Fig. 46. 



To see how the force P arises we consider the motion of the driving wheel 

 and the motion of a wheel of one of the coaches of the train. The left-hand 

 circle represents the coach-wheel, and the right-hand circle the driving wheel. 

 At starting the centre of the driving wheel is at rest, and the machinery is so 

 contrived that a couple is exerted upon it tending to turn it rapidly in the direc- 

 tion shown by the arrow-head marked CD. The wheel slips (or "skids") on the 

 rail, and, at the point of contact, friction is exerted in the direction opposite to 

 that in which the point of contact slips, this is shown by F in the figure. So 

 long as the steam is "on" the couple is exerted on the wheel, and there is slid- 

 ing friction F as shown. The resultant of such frictions for all parts on 

 which they act is the force P. Again, consider the coach-wheel rolling on the 

 rail with an angular velocity in the direction shown by the arrow marked co'. 

 The motion of the coach on the rails is in the direction of the arrow 

 marked V. In the absence of friction the coach would slip over the rails in 

 this direction. Thus the friction of the rails on the coach-wheels is in the 



