270 BOOK OP MATHEMATICAL PROBLEMS. 



A, P simultaneously ; prove that the least distance between them 

 during the motion is equal to the distance of P from AB. 



1290. A number of heavy particles start at once from the 

 vertex of an oblique circular cone whose base is horizontal, and 

 fall down generating lines of the cone ; prove that at any sub- 

 sequent moment they will lie in a subcontrary section. 



1291. The locus of a point P, such that the times of fall- 

 ing down PA, PB to two given points A, B may be equal, is a 

 rectangular hyperbola. 



1292. The locus of a point P, such that the time of falling 

 down PA to a given point A is equal to the time of falling verti- 

 cally from P to a given straight line, is one branch of a hyperbola 

 of which one asymptote is vertical, and the other perpendicular 

 to the given straight line. 



1293. A parabola is placed with its axis vertical and vertex 

 downwards ; prove that the time of falling down any chord 

 to the vertex is equal to the time of falling vertically through 

 a space equal to the parallel focal chord. 



1294. An ellipse is placed with its major axis vertical; 

 prove that the time of descent down any chord to the lower 

 vertex, or from the higher vertex, is proportional to the length of 

 the parallel diameter. 



1295. Two weights TF, nW move on two inclined planes and 

 are connected by a fine string passing over the common vertex, 

 the whole motion being in one plane ; prove that the centre of 

 gravity of the weights describes a straight line with uniform 

 acceleration 



n sin 8 sin o - 

 9 (n + l) a ^ w +2n cos (a + 



a, P being the angles of inclination of the planes. 



