CALCULUS. :M7 



1040. Of all acute angled triangles inscnl>rd in a given circle 

 of radius a, the mean value of the perimeter is r . 



1041. The mean value of the distance from one of the foci of 

 all points within a given prolate spheroid is - , 2a being 

 the major axis and e the eccentricity. 



1042. If a rod of length a be marked at random in two 

 points and divided at those points, the im-an value of the sum 



of the squares on the parts is - : and if the rod be first divided at 



i 



in into two parts, and the larger part again divided at 

 random, the mean value of the sum of the squares on the three 



. 35 , 

 parts IB ^a f . 



1043. If a, /?, y be the areal coordinates of a point, the mean 

 value of >/a/3y for all points within the triangle of reference 



l- 



h 105' 



1044. In the equation aj* qx + r = 0, it is known that 7 and 

 r l>oth lie between +1 and -1; assuming all values bet\\-i-n 



these limits to be equally pmli.-ililr, pnvi- that the chance of all 



. 

 ote of the equation being real i- 



1045. If a given finite straight line be di\ :id.m in 

 two points, the chance that tin- three parts can be sides of an 

 acute angled triangle is 3 1 



104G. If a rod be div 1 1 at r.u..l..in in t \vo points, and i 

 a chance that n times th. the squares on th- 



is less than the square on the whole line, prove that 



