218 BOOK OF MATHEMATICAL PROBLEMS. 



1047. If n points be taken at random on a given finite 

 straight line, the chance that one of the n+ 1 parts into which the 



rod will be divided shall be greater than half the line is ^- . 



1048. If a rod be divided at random into four parts, the 

 chance that one of the parts shall be greater than half the rod 



is ^; and the chance that three times the sum of the squares 

 on the parts shall be less than the square on the whole line 



7T 



is -. 



1049. If a given finite straight line of length a be divided at 

 random in two points, the chance that the product of the three 



parts shall exceed - , is 



Sir 



\ r si 



y J^if 



1050. If a given finite straight line be divided at random in 

 (1) four points, (2) n points, the chance that (1) four times, (2) n 

 times the sum of the squares on the parts shall be less than the 

 square on the whole line is 



1051. Find the singular solution of the equation 

 dy 



and trace the locus of the points whose coordinates satisfy the 

 singular solution. 



1052. Find the differential equation of a curve, such that 

 the foot of the perpendicular from a fixed point on the tangent 



