1 12 CEYI 1:1: "i- i,i:vvn Y. 



(8) 'n are placed at the corners of <i f> fi-nl> 



mass of each pat '/ t<> tin <n->n f tin 



opposite face : shew that the centre of gravity /stem 



><les with the centre of the spfiere i 

 hedron. 



Let A, B, C, D be tin- angular 

 Let p be the perpend it -ular from D on the face ABO* 

 Thru the distance of the centre of gravity of the system from 

 plane ABC 



x area of face 



~~ sum of the areas of the faces 



3 x volume of tetrahedron 

 ~" sum of the areas of the faces " 



And this expression is equal to the radius of the sphere 

 inscribed in the tetrahedron. 



:ice the required result is obtained. 



(9) A polyhedron is cir< > <l J></t a spJtrrr ; at ////> 



Efl of contact masses are placed which are proportional to tin- 



areas of the corresponding faces of the ]><>(>///// tl<<it 



the centre of gravity of these masses coincides ?///// //// centre 



of the sphere. 



Take the centre of the sphere for origin, and any plane 

 through the origin for the plane of (x, y). 



I ! ., A, A % , ...... denote the areas of the faces of the 



polyhedron; let z^z , z s ,... denote the ordinates of the points 

 the orainate of the centre of gravity. Then, by 



;e, 



-^_A l 

 Now the projection of the area A l on the plane of (x, y) 



A p 



- 1 - 1 , where r x is the radius of the sphere; and similarly 

 *j 

 for the other projections. And the sum of such projections 



is zero. Thus ~z =0; and since the plane of (x, y) is any 

 plane through the centre of the sphere, the centre of gravity 

 must coincide with the centre of the sphere. 



