!6 INTRODUCTION TO DYNAMICS. [34. 



must lie. The intersection of these lines gives the centroid of 

 the quadrilateral. 



34. For some purposes it is convenient to find a system of 

 particles whose centroid shall be the same as that of a quadrilat- 

 eral. The problem is of course indeterminate and may be 

 solved in various ways. 



Let m be the mass of the quadrilateral ABCD ; m lt m% the 

 masses of the triangles ABC, ADC. By Art. 32, each of these 

 triangles can be replaced by three equal particles -J^, \^n^ 

 placed at the vertices. We thus have at A, as well as at C, a 

 mass | (*#! 4- m^ = ^m. 



The masses ^m 1 at B and ^m 2 at D, whose sum is also 

 = \m, are proportional to the areas of the triangles ABC, ADC, 

 or to the lengths EB, ED, if E be the intersection of the 

 diagonals. Now these two different masses at B and D can be 

 replaced by a system of three masses, \m at B, \m at D, and 

 \m at E. For (i) the total mass evidently remains the same, 

 and (2) the centroids of the two systems coincide as is easily 

 seen by taking moments with respect to E. 



Indeed, the centroid G' of ^m l at B and ^m z at D is deter- 

 mined by the equation 



(m l 4- m 2 ) EG' = m 1 EB m^ ED ; 



substituting for m lt m% their values as found from the relations 

 m l + m 2 = m, m l /m <2i = EB/ED, this reduces to 



m>EG' = m-(EB-ED). 



The centroid G u of \m at B,\m at D, and \m at E is 

 given by 



Hence G' and G" coincide. 



The centroid of the area of a homogeneous quadrilateral is 

 therefore the same as that of four equal particles placed at its 



