42 CENTRE OF GRAVITY OF MIXED SPACE 



which being multiplied by its length, gives 



2352 X 50=117600; 

 consequently, by subtraction, the dividend becomes 



117600 53760 = 63840. 



Now, the second clause of the second rule, being the same as the 

 second clause of the first rule, it follows, that the divisor must here 

 be the same, as we have found it to be in the preceding case ; conse- 

 quently, by division, we obtain 



therefore, from the numerical values of the co-ordinates as we have 

 just determined them, the position of the centre of gravity of the 

 proposed figure can easily be found, in the following manner. 



57. Let ABCD represent the rectangular parallelo- 

 gram, of which the side AB is 28 feet, and the side EC 

 50 feet; and let EDC be the right angled triangle, 

 whose perpendicular E D is 42 feet, and its base D F 20 

 feet, all taken from the same scale of equal parts. 



From the angle B, and on the sides BC and BA, set 

 off BS and Br respectively equal to 20.286 and 10.857 

 feet; then, through the points s and r, draw the lines sn and rn, 

 respectively parallel to AB and BC, and the point n is the Centre of 

 gravity of the figure ABCFE, which remains after the right angled 

 triangle EDF is separated from the parallelogram ABCD. 



If the line of division, or hypothenuse of the triangle EF, were 

 parallel to AC the diagonal of the parallelogram, as is distinctly speci- 

 fied in the foregoing problem, the solution would become much more 

 simple ; for then, in order to determine the position of the centre of 

 gravity, it is only necessary to reduce one of the equations, and it is 

 altogether a matter of indifference which of them it is, provided that 

 the conditions of the equation be strictly attended to. 



Supposing E D the perpendicular of the triangle, to remain as above ; 

 then the base, when the hypothenuse is parallel to the diagonal of the 

 rectangle, will be found by the following analogy, viz. 

 50 : 28 : : 42 : 23.52. 



Then, by calculating according to rule first, or equation (20), our 

 dividend and divisor are 103313.28 and 5436.48 respectively; con- 

 sequently, we get 



103313.28 



*= 5436.48 = 19fe 



