. Spons' dictionary of engineering, civil, mechanical, military, and naval; with technical terms in French, German, Italian, and Spanish . termination of the centreof gravity of the area included under a closed curve the equationof which is given. If we have to consider an area enclosed by an irregularcurve, or one whose equation we do not possess, we may makeuse of the following method ;—Draw lines parallel to the 3329, at equal distances apart, these distances being smallenough to allow us to consider the portions of the contour includedbetween two consecutive parallels as sensibly
. Spons' dictionary of engineering, civil, mechanical, military, and naval; with technical terms in French, German, Italian, and Spanish . termination of the centreof gravity of the area included under a closed curve the equationof which is given. If we have to consider an area enclosed by an irregularcurve, or one whose equation we do not possess, we may makeuse of the following method ;—Draw lines parallel to the 3329, at equal distances apart, these distances being smallenough to allow us to consider the portions of the contour includedbetween two consecutive parallels as sensibly rectilinear. Thewhole figure is thus divided into squares, rectangular trapeziumsand rectangular triangles, figures whose areas and centres ofgravity we are able to determine. Taking the moments of thesepartial areas with respect to the two axes and summing, we getthe moments of the total area ; and as this total area is the sum of the partial areas, by dividingthe moments found by this total area, we obtain the co-ordinates of the centre of gravity with anapproximation that increases as the space between the parallels 1710 GEAVITY. VI. We have now to determine the centre of gravity of curved surfaces. Surfaces of Revolution.—Let O X, Fig. 3327, be the axis of revolution, and A B the generatingline, or generatrix, whose equation is supposed to be given ; and let O 0 = a and O D = 6 be theabscissae of the planes perpendicular to the axis O X serving as limits to the surface. Divide thissurface by planes M P, M P, perpendicular to the axis of revolution, into infinitely small zones,which may be considered as surfaces of frusta of cones. Let x and y be the co-ordinates of thepoint M, and s the arc A M of the generatrix. We shall have as the expression of the surface of the frustum generated by the element MM or d s, - / 2 tt?/ + 2 tt (?/ + d^/) ) £?s or 2tt?/ ds, by neglecting the infinitely small d y before the finite quantity y. The centre of gravity of this el
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