EXAMPLES. 79 



91. A particle is describing a circle of radius c as a central orbit about a 

 point distant c/v/3 from the centre. When the line joining this point to the 

 particle subtends a right angle at the centre of the circle the law of the 

 acceleration suddenly changes, and thereafter it varies inversely as the square 

 of the distance, but the magnitude of the acceleration does not change 

 discontinuously. Prove that the major axis of the new elliptic orbit is 

 lGc/5^/3 and that its eccentricity is J19/8. 



92. Prove that the focal radius and vectorial angle of a particle describing 

 an ellipse of small eccentricity e at time t after passing the nearer apse are 

 approximately given by the equations 



r = a(\ -ecosnt+%e 2 -$e 2 cos2nt), 



= nt + 2e sin nt + e 2 sin 2nt, 

 where 2a is the major axis and 2ir/n is the periodic time. 



Prove also that if e 2 is neglected the angular velocity about the other focus 

 is constant. 



93. Prove that the time of describing the smaller part of an elliptic orbit 

 about a focus cut off by a focal chord is v /( a3 //^) (20 sin 20), where 2a sin < is 

 the chord of the auxiliary circle that corresponds to the focal chord, and 2a is 

 the major axis of the orbit. 



94. If the perihelion distance of a comet is - th of the radius of the 



earth's orbit, supposed circular, show that the comet will remain within the 

 earth's orbit for 



years, 

 on 



the comet's orbit being parabolic. 



95. If the parabolic orbits of two comets intersect the orbit of the earth, 

 supposed circular, in the same two points, and if t lt t z are the times in which 

 the comets move from one of these points to the other, prove that 



(<i+<iJ l +(<i-<i) l -(g; T}* > where T is a year. 



96. The times of passage of a particle between two points distant d apart 

 in the two parabolic orbits that can be described about the same focus with 

 the same law of acceleration are T lt T 2 , and the distances of the points from 

 the focus are r lt r 2 . Prove that 



97. Three focal radii SP, SQ, SR of an elliptic orbit about a focus S are 

 determined, and the angles between them. Show that the ellipticity may be 

 found from the equation >A = aA', where A is the area of the triangle PQR, 

 and A' is the area of a triangle whose sides are 



and two similar expressions. 



