OF IMMERSED RECTILINEAR FIGURES, 41 



remainder by twice its area ; then, subtract the product from 

 three times the square of the breadth of the parallelogram 

 drawn into its length, and the remainder will be the dividend. 

 Divide the dividend above determined, by three times the 

 difference between twice the area of the parallelogram, and 

 twice that of the triangle, and the quotient will give the 

 co-ordinate of the line BC. 



56. EXAMPLE 9. The sides of a rectangular parallelogram are 

 respectively 28 and 50 feet, and from one of the lower corners, is 

 separated a right angled triangle, by means of a straight line ter- 

 minating in the adjacent sides; it is required to determine the 

 position of the centre of gravity of the remaining part, the base and 

 perpendicular of the separated triangle, being respectively equal to 

 20 and 42 feet ? 



Here then, by operating as directed in the first rule, we have 



3 x 50 42 = 150 42 = 108, 

 and by the principles of mensuration, twice the area of the triangle, is 



42 X 20 = 840 square feet ; 

 therefore, by multiplication, we obtain 



108 x 840 = 90720. 

 Again, three times the square of the length of the parallelogram, is 



3x 50 2 = 7500, 

 which being multiplied by its breadth, gives 



7500 x 28 = 210000; 

 consequently, by subtraction, the dividend is 



210000 90720=119280. 



Now, twice the area of the parallelogram, is 2 x 50 x 28 = 2800 

 square feet, and twice the area of the triangle, is 42 x 20 = 840 

 square feet ; therefore, by the second clause of the rule, we obtain 



a = H928 = 20.286 feet nearly. 



3(2800 840) 



Hence it appears, that the co-ordinate of the line AB, according to 

 the proposed data, is very nearly 20.286 feet; and by operating as 

 directed in the second rule, we shall have 



3 x 28 20 = 84 20 = 64, 

 and by the principles of mensuration, twice the area of the triangle, is 



42 X 20 = 840 square feet ; 

 therefore, by multiplication, we obtain 



64 x 840 = 53760. 



Again, three times the square of the breadth of the parallelogram, is 

 3 x 28* = 2352, 



