258 BOOK OF MATHEMATICAL PROBLEMS. 



1239. The limiting position of the centre of gravity of the 

 area included between the area of a quadrant of an ellipse bounded 

 by the axes and the corresponding quadrant of the auxiliary circle, 

 as the ellipse approaches the circle as its limit, will be a point 

 whose distance from the major axis is twice its distance from the 

 minor axis. 



1 240. A curve is divided symmetrically by the axis of #, and 

 is such that the centre of gravity of the area included between 

 the ordinates x = 0, x = h is at a distance mk from the origin ; 

 prove that the equation of the curve is 



SfTV-l 



y = Cx 1 -. 



1241. The circle is the only curve in which the centre of 

 gravity of the area included between any two radii vectores and 

 the curve lies on the straight line bisecting the angle between the 

 radii. 



1242. Determine the differential equation of a curve such 

 that the centre of gravity of any arc measured from a fixed point 

 lies on the straight line bisecting the angle between the radii of 

 the extremities. Prove that the curve is a lemniscate, the node 

 being pole. 



1243. Two rods AB, BC rigidly united at B and suspend- 

 ed freely from A, rest inclined at angles a, /? to the vertical; 

 prove that 



1244. AB, are two uniform rods freely jointed at B and 

 moveable about A which is fixed ; find at what point in BG a 

 prop must be placed so that the rods may be at rest in a hori- 

 zontal straight line. 



1245. Three equal uniform rods, jointed together at their 

 extremities, rest in one horizontal line on three pegs, each rod in 

 9ontact with one peg; find the positions of equilibrium. 



