EXAMPLES. 83 



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

 orbits about with accelerations /, / , prove that 



**f,r&amp;gt;3f = 2 



h 2 h 2 sin* &amp;lt; 



where h and h are constants, r and r are corresponding radii vectores, and 

 &amp;lt; 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 



where u is the reciprocal of the distance from 0. 



128. If the central acceleration is p, [2 (a 2 + b 2 } u 5 - 3a 2 W], the initial 

 distance a, and the initial velocity ij^ja at right angles to the radius vector, 

 determine the orbit. 



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

 tion pu 3 (n 2 + l-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=acoshnd. 



130. A particle describes a central orbit with acceleration 



starting from a point for which r = a with velocity 3 ^(2a/n) in a direction 

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



131. If the central acceleration is 2/z (u 3 - a 2 u 6 ) and the particle is pro 

 jected from an apse at distance a with velocity vW a &amp;gt; the time until the 

 distance is r is 



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

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

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

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



133. A particle describes a central orbit with acceleration pr/(r 2 a 2 ) 2 

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

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

 if the equations of the cardioid are 



x 2a cos - a cos 20 



y = 2a sin &amp;lt; - a sin 20 



the time between two points &amp;lt;/&amp;gt;j and &amp;lt;p 2 is the same as in an elliptic orbit 

 about a focus between two points whose eccentric angles are 0j and 2 , 

 the periodic times being the same. 



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