68 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



whatever the number we agree to neglect may be. In this case 

 the orbit tends to depart widely from the circular form. 



In the former of these cases the circular motion is said to be 

 stable, in the latter unstable. 



66. Examples. 



1. If f(r}=r~ n or <f>(u) = u n , prove that the possible circular orbits are 

 stable when n<% and unstable when n>3. 



2. For n=3 prove that the circular orbit is unstable, and find the orbit 

 described by a point moving with the h for circular motion in a circle of radius 

 c through a point near the circle. 



3. If /"(r)=r~ 4 prove that the curve described with the h for circular 

 motion in a circle of radius c is either the circle r = c or one of the curves 



r cosh + 1 r_ cosh 6 1 

 ~c = coshT-~2 ' c ~ cosh + 2 ' 



EXAMPLES. 



1. Prove that the time in which it is possible to cross a road of breadth c, 

 in a straight line, with the least uniform velocity, between a stream of 

 omnibuses of breadth b, following at intervals a, moving with velocity F, is 



b 



2. A particle moves in the plane of two rectangular axes so that the 

 resolved parts of its velocity parallel to the axes are proportional to its 

 distances from two other rectangular lines in the plane. Prove that its path 

 is an equiangular spiral or a hyperbolic spiral. 



3. Three horses in a field are at a certain moment at the angular points 

 of an equilateral triangle. Their motion relatively to a person driving along 

 a road is in direction round the sides of the triangle (in the same sense) and 

 in magnitude equal to the velocity of the carriage. Show that the three 

 horses are moving along concurrent lines. 



4. A straight line AB turns with uniform angular velocity about a point 

 A, retaining a constant length, and a second straight line EG of constant 

 length moves so that C is always in a certain straight line through A. Prove 

 that the velocity of C is proportional to the intercept which/?<7 makes on the 

 line through A at right angles to AC. 



5. A point P moves uniformly in a circle ; Q is a point in the same 

 radius at double the distance from the centre ; PR is a tangent at P equal to 

 the arc described by P from the beginning of the motion ; show that the 

 acceleration of R is represented in direction and magnitude by RQ. 



