280 BOOK OF MATHEMATICAL PROBLEMS. 



i:U5. At any point in an elliptic orbit about the focus, the 

 velocity v receives a small increment Sv; prove that the alterations 

 in the eccentricity e t and the longitude of the apse i cr, will be 

 given by the equations 



&j efrsr 



6* (a - r) ab ^{aV - (r - a)*} p.ae (2a - r) ' 



1346. An ellipse is described by a particle under the action 

 of two forces tending to the foci, each varying inversely as the 

 square of the distance ; prove that 



2 a 5 (^w 2 + /n'w' 2 ) (<o -f to 7 )* 



a, b being the axes, and GO, <*>' the angular velocities at any point 

 about the foci. 



1347. Two fixed points of a lamina slide along two straight 

 lines fixed in space, so that the angular velocity of the line 

 joining the points is constant ; prove that (1) every fixed point of 

 the lamina describes an ellipse under acceleration tending to the 

 intersection of the two fixed straight lines and proportional to the 

 distance : (2) every fixed straight line of the lamina envelopes 

 during its motion an involute of a four-cusped hypocycloid : 



(3) the motion of the lamina may be completely represented by 

 supposing a circle fixed in the lamina to roll uniformly with 

 internal contact on a circle of twice its radius fixed in space : 



(4) for a series of points of the lamina lying in a straight line, the 

 foci of the ellipses described lie on a rectangular hyperbola. 



1348. If a lamina move in its plane so that two fixed points 

 in it describe straight lines with acceleration/,/'; the accelera- 

 tion of the centre of instantaneous rotation is 



1 

 sin 



being the angle between the lines. 



