Centre of Gravity in particular Bodies. 67 



i IE X (DF + 2 TH) m 94 



Z>jF + TH 



that of the trapezoid TKMH from the same point , will be, for 

 the same reason, taken in connexion with the equality of the 

 lines IE, 7L, &c., 





 TH + KM TH + JSTJtf 



In like manner, the distance of the centre of gravity of the 



trapezoid NKMP, will be 



, * ,p , 



+ AP J5CJH + AP 



and so on. 



Now if we multiply each distance by the surface of the cor- 

 responding trapezoid, that is, by half the sum of the two parallel Ge om. 

 sides, multiplied by their common height IE, we shall have for 178 - 

 the series of these moments, 



J IE 3 X (DF + 2 TH\ J l 3 X (4 TH + 5 ^Jf), 



J Ifi 3 X (7 KM + 8 A"P), 

 and so on ; the sum of which will be 



J IE* X (JDF -f 6 TH -f 12 JO/ + 18 JVP -f 

 24 QS 4- 14 



It may be observed, that if there were a greater number 

 of divisions, the multiplier of the last term, which is here 14, 

 would be in general 2 -f- 3 (n* 2) or 3 n 4, n represent- 



ing the whole number of the perpendiculars DjP, TH, &c., in- 

 cluding AB, which may be zero. So that the general expression 

 for the sum of the moments, is reduced to 



IE 2 (J DF + TH + 2 KM -f 3 JVP -f 4 QS + &c. . . 



But it is evident that the surface ANDFPB has for its ex- 

 pression, 



IE X (J DF -f TH + ITJIf + JVP 4- & c . . . + \ AB) ; 

 and hence the distance of the centre of gravity G, namely, 



