. An elementary treatise of mechanical philosophy, wirtten for the use of the undergraduate students of the University of Dublin. ch, putting s for the generated surface, is s zz o). i. e. the generated surface is equal to the product of the ge-nerating line, into the arc or path described by its centre ofgravity. To prove the proposition for a solid of revolution, letCBDE be the generating plane, and cb, as before, the axis ofrevolution. The plane being supposed to be resolved intoelementary rectangles, by ordinates perpendicular to theaxis CB; and dx, denoting the portion of the axis bet
. An elementary treatise of mechanical philosophy, wirtten for the use of the undergraduate students of the University of Dublin. ch, putting s for the generated surface, is s zz o). i. e. the generated surface is equal to the product of the ge-nerating line, into the arc or path described by its centre ofgravity. To prove the proposition for a solid of revolution, letCBDE be the generating plane, and cb, as before, the axis ofrevolution. The plane being supposed to be resolved intoelementary rectangles, by ordinates perpendicular to theaxis CB; and dx, denoting the portion of the axis betweenany two consecutive ordinates, the expression for the ele-mentary rectangle will be //.dx. But the sum of the products had by multiplying each of OF THE CENTRE OF GRAVITY. 119 these elementary rectangles, into the distance of its centre ofgravity from the axis, is equal to the single product of thearea of the entire plane, into the distance of its centre of gra-vity from the same axis, i. e. putting a for the entire areaCBED, and g for the distance of its centre of gravity from theaxis CB, it will be ,2 S Y ^^^ - ^•^^. Therefore, The first member of this equation is the sum of the pro-ducts of into -~. or the sum of the products had by multiplying each elementary rectangle into half the arc de-scribed by the extremity of its ordinate, i, e. the sum of theportions of the solid generated by the several rectangles :and the second member is the area of the generating plane,multiplied into the arc or path described by its centre ofgravity. Wherefore, putting v for the entire volume of thegenerated solid, it will be V ^ A.( The whole of the surface or solid of revolution is ex-pressed by replacing w in these formulae with 2 vr, the ab-stract number, which denotes the ratio of the periphery of acircle to its radius, i. e. the quotient of the former dividedby the latter. Making this substitution, the surface of re-volution is expressed by the equation s z: t
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Keywords: ., bookcentury1800, bookide00leme, booksubjectdynamics, bookyear1835
