BOOK OF MATHEMATICAL PROBLEMS. 



m*a 



new orbit will be - = , 4a being the latus rectum of the 

 parabola, and m the root of the equation 





which lies between 1 and e. 



1402. In an orbit described under a central force a straight 

 line is drawn from a fixed point perpendicular to the tangent and 

 proportional to the force, and this straight line describes equal 

 areas in equal times ; prove that the equation of the orbit is 

 of the form 



Prove that the rectangular hyperbola is a particular case. 



1403. A chain of uniform material rests under normal and 

 tangential forces n, t; prove that the curve in which it rests 

 could be described by a particle, whose mass is equal to that of 

 a unit of length of the chain, under the action of normal and 

 tangential forces 2n, t. 



1404. A centre of force varying inversely as the w th power of 

 the distance moves in the circumference of a circle, and a particle 

 describes an arc of the same circle under the action of the force ; 

 prove that the velocity of the centre of force : velocity of the 

 particle :: 5 n : 1 n. If n = 3 t the time of describing a 



/2 

 semicircle is 4a 8 / - . 



V /* 



1405. A particle P is repelled from a fixed point S by a force 

 varying as (dist.)~ a , and attracts another particle Q with a force 

 varying as (dist.)" 8 ; initially P and Q are equidistant from S in 

 opposite directions, P is at rest, and the accelerations of the two 

 forces are equal ; prove that if Q be projected at right angles to 

 8Q with proper velocity, it will describe a parabola of which S is 

 the focus. 



