EXAMPLES. 201 



EXAMPLES. 



1. Prove that the time of quickest descent along a straight line from 

 a point on one vertical circle to another in the same plane is 



where c is the distance between their centres, a is the sum of the radii, and h 

 the vertical height of the centre of the former circle above that of the latter. 



2. Any curve is drawn in a vertical plane, and a second curve is drawn 

 cutting off equal distances along the normals to the first curve. Prove that, 

 if the second curve now receives a certain vertical displacement, the time of 

 quickest descent from one curve to the other is independent of the starting 

 point. 



3. Show that when two curves lie in the same vertical plane and do not 

 intersect, the straight line of quickest descent from one to the other is such 

 that the normals and the vertical lines through its extremities form a 

 rhombus ; and further that the centres of curvature at the extremities cannot 

 lie on the segments of the normals included between the verticals. 



4. A parabola is placed in a vertical plane with its axis inclined to the 

 vertical at an angle cos" 1 f , the vertex being the highest point of the axis. 

 Prove that the time of sliding down the latus rectum is the same as that 

 of sliding down the chord drawn from the upper end of the latus rectum to 

 the vertex, and that the time down any intermediate chord is less. 



5. A parabola is placed in a vertical plane with its vertex downwards and 

 its axis inclined to the vertical at an angle . Prove that the time down the 

 chord of quickest descent from the focus to the curve is v /(2a.9 r ~ 1 se 



6. A spherical shell has a small hole at its lowest point, and any number 

 of particles start down chords from the interior surface at the same instant, 

 pass through the hole, and then move freely. Prove that at any instant 

 before or after passing through the hole they lie on the surface of a sphere, 

 and find the radius and position of such a sphere. 



7. An ellipse is placed with its minor axis vertical. Prove that the 

 normal chord of quickest descent from the curve to the major axis is that 

 drawn from a point subtending a right angle at the foci when there is such a 

 point. Determine the normal chord of quickest descent when there is no 

 such point. 



