EXAMPLES OF MAXIMA AND MINIMA. 247 



24. Determine a point within a triangle such that the sum 



of the squares on the distances from the three angles 

 is a minimum. 



Result. The centre of gravity of the triangle. 



25. Through 'a point within a triangle three straight 



lines are drawn parallel to the sides dividing the 

 triangle into three parallelograms and three triangles : 

 shew that the sum of these triangles is least when the 

 straight lines are drawn through the centre of gravity 

 of the triangle. 



26. A triangular space is to be diminished by fencing off 



the corners, each fence being circular and having the 

 nearest corner as centre: shew how to leave the 

 greatest possible central space with a given length of 

 fence. 



Result. The radii of the circular fences are equal. 



27. Given the sum of the three edges of a rectangular 



parallelepiped, find its form that its surface may be 

 a maximum. 



28. In a given sphere inscribe a rectangular parallelepiped 



whose volume is a maximum. Also one whose surface 

 is a maximum. 



Result. A cube. 



29. Of all triangles of the same perimeter find that which 



will generate the greatest double cone by revolving 

 about a side. 



Result. The fixed side must be two-thirds of each of 

 the other sides of the triangle. 



3'J. A rectangular parallelepiped is so constructed that a 

 plane which passes through three of its corners, but 

 through no edge, contains a point whose distances 

 from the three faces adjacent to one of the other 

 corners are given. Shew that the shortest diagonal 

 which such a parallelepiped can have, is (a^ + $ + c^)*> 

 where a, b, c are the given distances. 



