EXAMPLES. 83 



126. If inverse curves with respect to can be described as central 

 orbits about with accelerations /, /', prove that 



fff_ L ry_ 2 



h z """ h 1 * ~ sin* ' 



where 7i and h' are constants, r and r' are corresponding radii vectores, and 

 $ is the angle r or / makes with the tangent. 



127. If /is the acceleration and %h the areal velocity in a central orbit 

 about a point 0, prove that the angular acceleration a about satisfies 

 the equation 



_ 

 du u 



where u is the reciprocal of the distance from 0. 



128. If the central acceleration is p. [2 (a 2 + b 2 ) u*> - 3a 2 b V], the initial 

 distance a, and the initial velocity *Jn/a at right angles to the radius vector, 

 determine the orbit. 



129. A particle describes a central orbit about the origin with accelera- 

 tion ^u 3 (n 2 + 1 - 2n 2 a 2 u 2 ), starting from an apse at distance a with the 

 velocity from infinity ; prove that it describes the curve 



r=acoshn0. 



130. A particle describes a central orbit with acceleration 



p [4 (a/r) 9 + (a/r) 3 - 32 (r/a) 3 ] 



starting from a point for which ra with velocity 3 ^(%a\i) in a direction 

 making an angle TT with the radius vector. Prove that the path is 



131. If the central acceleration is 2p (u 3 - a?u 5 ) and the particle is pro- 

 jected from an apse at distance a with velocity vW a ? ^ ne time until the 

 distance is r is 



132. A particle moving with a central acceleration /x(w 4 + 2aw 5 ) starts 

 from a point at distance a from the origin in a direction making an angle 

 (TT - cot" 1 2) with the radius vector and with the velocity from infinity. Show 

 that the equation of the path is ra (1 - 2 sin 0). 



133. A particle describes a central orbit with acceleration p.r/(r 2 a 2 } 2 

 towards the origin being projected from an apse at distance 3a with velocity 

 /^(2/*)/4a. Show that it describes the cardioid r 2 =a 2 + fp 2 . Prove also that 

 if the equations of the cardioid are 



y = 2a sin $ - a sin 20 



the time between two points <p 1 and < 2 is the same as in an elliptic orbit 

 about a focus between two points whose eccentric angles are <j and </> 2 , 

 the periodic times being the same. 



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