STATICS. 257 



IL Centre of Gravity. 



1234. A rectangular board of weight IT is supported in a 

 horizontal position by vertical strings at three of its angular 

 points ; a weight 5 W being placed on the board, the tensions of 

 the strings become W, 2 IF, 3ir ; prove that the weight must be at 

 one of the angular points of a hexagon whose opposite sides are equal 

 and parallel, and whose area is to that of the board as 3 : 25. 



1235. If particles be placed at the angular points of a tetra- 

 hedron, proportional respectively to the areas of the opposite 

 faces, their centre of gravity will be the centre of the sphere 

 inscribed in the tetrahedron. 



1236. A uniform wire is bent into the form of three sides of 

 a polygon AB, BC, CD, and AB**CD = a, BC=b', prove that, 

 if the centre of gravity of the wire be at the intersection of AC, 

 1>1>, each of the angles B t G is equal to 



-(-) 



1237. A thin uniform wire is bent into the form of a triangle 

 ABC, and particles of weights P, Q, R are placed at the angular 

 points; prove that, if the centre of gravity of the particles coin- 

 cide with that of tin- v. 



P : Q i R :: b + c : c + a : a + b. 



1238. A polygon is such that a,, a f , a,... being the angles 

 made by its sides with any fixed straight line, 



2(cos2a)0, 2(sin2a) = 0; 



prove th:it there exists a point whirh is tin' centre of gra\ i 

 n r.junl ]>.ir tides placed perpendiculars from 



on the sides; and that the centre of gravity of n eq>: 

 the feet of the perpendiculars from any other point 1\ bisects OP t 

 w. 17 



