Ill 



CENTHE OF GRAY II T. 



|>nneij.le is the s; body 18 divided into an indefmitelv 



large number of indefinitely small element^ : tin- volin 

 an element is estimated, and this IK-HILT multiplied by the 

 density gives the mass of the element. Tin- nia.-s i- multi- 

 plied I))' the abscissa of the element, and we lind tin- Mini 

 of the values ;>roduct for all tl; 



corresponds to the ^.Px of Art. (\(\. Also we find the 

 of the masses of all the . and thus obtain a result 



to the ^7* of the sain . Ih'vide the 



former r-sult ly the latter and we have the va! 

 similarly // and .: eau be found. In the following examples 

 the student must not allow the details of the ! Cal- 



culus to obscure his recognition of the fundamental fnimula 

 of Art. 6G ; he must consider in e\ what convsjMMids 



to the P, x, y, z of that Article, that is, he must carefully as- 

 in into what elements the body is decomposed. 



ordinates 



Plane Area. 



111. Let CBEU be an area bounded 

 BG and 7^77, the curve 

 BE, and the portion CJf 

 of the axis of x ; it is re- 

 ijuired to find the centre 

 of gravity of the area. Or 

 instead of the area we 

 may ask for the centre of 

 gravity of a solid bounded 

 by two planes parallel to 

 th' plane of the paper and equidistant from it, and by a str 

 line which moves round the boundary CBJ-1II remain ii 

 ways perpendicular to the plane of the paper. Divide Cll into 

 n portions, and suppose ordinates drawn at the joints of divi- 



1 MQ represent two consecutive nrdi: 

 and draw PN parallel to LM. 



= c, OH=h. 



The area of the rectangle PM is yA#; suppose u to denote 

 the area of PQX, and let x be the abscissa of the centre of 



