APPLICATION AND DEVELOPMENT OF PRISMOIDAL FORMULA. 141 



14. Centre of gravity of various sections. — If the line perpen- 

 dicular to the axis of rotation, by which a solid is generated, be 

 taken as the axis of abscissae, we shall require only the abscissa 

 of the centre of gravity of each section previously discussed. The 

 triangular section shewn in Fig. 3 is first considered. Let the 

 line FG be bisected in M; then since the centre of gravity of the 

 triangle DFG is on the line DM, and at the point O so taken that 

 DO = 2*OM ; or what is the same thing, since the abscissa of O is 

 | that of M, reckoned from the axis DE ; or still more generally 

 since the coordinates of are the mean of those of the three points 

 of the triangle : 



*°=l* c ^ ( 28 > 



as may easily be verified, see (14). If the perpendicular distances 

 d and d' of F and G- from the line DE be given, then whether FE 

 and EG are in one line or not, the centre of gravity is one-third 

 their difference : that is 



x =l(d'-d) (29) 



the identity of which with (28) is easily shewn. The expression 

 for the centre of gravity of sections like that illustrated in Fig. 4 

 is not so simple. By equating the moments of the several triangles 

 composing the figure about the y axis, parallel to CD, it will be 

 seen that 



^ _ L d'e'^ + d')-dee§ + d) _ L E>a + d')-E(>i±d) 

 cw + d'e +de A 



E' and E denoting the areas of the pairs of triangles standing on 

 the lines e and e. respectively, and A the area of the whole figure. 

 If in Fig. 4, the centre of gravity be required for the whole figure 

 including the triangle ABJ, it is merely necessary to substitute 

 C for c in formula (30), and A must include this triangular area. 

 More generally, if J, A' etc., be contiguous areas the abscissa* of 

 whose centres of gravity are x, x etc. from any arbitrarily chosen 

 axis, then the abscissa of the centre of gravity of the total area 



~ Ais X ^^YT- < 31 > 



from which formula* for particular cases may be readily deduced. 



