C If the segment become a hemisphere, 2,=a, .G,=ža. Л 36. The centre of gravity of the sector of a circle. Let CAB represent such a sector; conceive the arc ADB to be a polygon of an infinite number of sides and lines, to be drawn from all the angles of the polygon to the centre C of the circle, these will divide the sector into as many triangles. Now the centre of gravity of each triangle will be at a distance from C equal to the line drawn from the vertex C of that triangle to the bisection of its base, that is equal to the radius of the circle, so that the centres of gravity of all the triangles will lie in a circular are FE, whose centre is C and its radius CF equal to CA, and the weights of the triangles may be supposed to be collected in this are FE, and to be uniformly distributed through it, so that the centre of gravity G of the whole sector CAB is the centre of gravity of the circular are FE. Therefore by equation (23), if S', C', and a', represent the are FE, its chord FE, and its radius CF, and S, C, a, the similar are, chord, and radius of a'C ADB, then CG = ; but since the arcs AB and FE are S' = similar, and that a' fa, . C' C and S'= S. Substi= a,:. S'S. tuting these values in the last equation, we have 37. The centre of gravity of any portion of a circular ring or of an arch of equal voussoirs. A C1 Let B,C,C,B, represent any such portion of a circular ring whose centre is A. Let a represent the radius, and C, the chord of the arc B,C,, and S, its length, and let a,, C, similarly represent the radius and chord of the arc B,C,, and S, Pthe length of that arc. G1 G Also let G, represent the centre of gravity of the sector AB,C,, G, that of the sector AB,C,, and G the centre of gravity of the ring. Then AG, xsect. AB,C,+AG × ring B,C,B,C,=AG, × sect. AB,C, Now (by equation 32), AG, C, AG,=† ;;; S2 also sector AB,C, 4S,a,, sector AB,C,S,a, = .. ring B,C,C,B,= sect. AB,C,- sect. AB,C,= 18, a, — 18,a, a2C2 ¿S,a,+ AG ‡ (S,a,—§,a,)=?“,, . §. §,a,, N S2 S, If NL represent any plane area, and AB be any axis, in the same plane, about which the area is made to revolve, so tha NL is by this revolution made to generate a solid of revolution, then is the volume of this solid equal to that of a prism whose base is NL, and whose height is equal to the length of the path which the centre of gravity G of the area NL is made to describe. M PQ B L For take any rectangular area PRSQ in NL, whose sides are respectively parallel and perpendicular to AB, and let MT be the mean distance of the points P and Q, or R and S, from AB. Now it is evident that in the revolution of NL about AB, PQ will describe a superficial ring. Suppose this to be represented by QFPK, let M be the centre of the ring, and let the arc subtended by the angle QMF at distance unity from M be represented by 8, then the area FQPK equals the sector FQM-the sector KPM MQ × 6- MP2 ×ê= K ¿ ¿ (MQ2 — MP®) =¿ (MQ + MP) 2 ×(MQ-MP)=8(MT × PQ). Now the solid ring generated by PRSQ is evidently equal to the superficial ring generated by PQ, multiplied by the distance PR. This solid ring equals therefore (MT × PQ × PR) or 8× MTX PRSQ. Now suppose the area PRSQ to be exceedingly small, and the whole area NL to be made up of such exceedingly small areas, and let them be represented by a,, a, a, &c. and their mean distances MT by a,, ,, &c. then the solid annuli generated by these areas respectively will (as we have shown), be represented by o x α, ôx ɑ ô x α, &c. &c.; and the sum of these annuli. 29 or the whole solid, will be represented by era, ++ ea,+&c., or by (c,a,+xa2+aa,+&c.). Now if a repre sent the weight of any superficial element of the plane NL, a=the moment of the weight of a, about the axis AB, that of the area a, about the same axis AB, and so on, therefore the sum (x,a,+x,а2+x ̧à ̧+&c.)μthe moment of the whole area NL about AB; but if G be the centre of gravity of NL, and GI its distance from AB, then the moment of NL about AB=GI×NL® ; therefore the whole solid=8. GI. NL; but . GI equals the length of the circu lar path described by G; therefore the volume of the solid equals NL multiplied by the length of the path described by G, i. e. it equals a prism NM, whose base is NL, and whose height GH is the length of the path described by G; which is the first property of GUL AL T B L N Ꮮ, N DINUS. 39. The above proposition is applicable to finding the solid contents of the thread of a screw of variable diameter, or of the material in a spiral staircase: for it is evident that the thread of a screw may be supposed to be made up of an infinite number of small solids of revolution, arranged one above another like the steps of a staircase, all of which (contained in one turn of the thread) might be made to slide along the axis, so that their surfaces should all lie in the same plane; in which case they would manifestly form one solid of revolution, such as that whose volume has been investigated. The principle is moreover applicable to determine the volume of any solid (however irregular may be its form otherwise), provided only that it may be conceived to be generated by the motion of a given plane area, perpendicular to a given curved line, which always passes through the same point in the plane. For it is evident that whatever point in this curved line the plane may at any instant be traversing, it may at that instant be conceived to be revolving about a certain fixed axis, passing through the centre of curvature of the curve at that point; and thus revolving about a fixed axis, it is generating for an instant a solid of revolution about that axis, the volume of which elementary solid of revolution is equal to the area of the plane multi plied by the length of the path described by its centre of gravity; and this being true of all such elementary solids, each being equal to the product of the plane by the corres ponding elementary path of the centre of gravity, it follows that the whole volume of the solid is equal to the product of the area by the whole length of the path. PTQ M B D 40. If AB represent any curved line made to revolve about the axis AD so as to generate the surface of revolution BAC, and G be the centre of gravity of this curved line, then is the area of this surface equal to the product of the length of the curved line AB, by the length of the path described by the point G, during the revolution of the curve about AD. This is the second property of Guldinus. C Let PQ be any small element of the generating curve, and PQFK a zone of the surface generated by this element, this zone may be considered as a portion of the surface of a cone whose apex is M, where the tangents to the curve at T and V, which are the middle points of PQ and FK, meet when produced. Let this band PQFK of the cone, QMF be developed, and let PQFK represent its development; this figure PQFK will evidently be a circular ring, whose centre is M; since the development of the whole cone is evidently a circular sector MQF whose centre M corresponds to the apex of the cone, and its radius MQ to the side MQ of the cone. F K Now, as was shown in the last proposition, the area of this circular ring when thus developed, and therefore of the conical band before it was developed, is represented by . MT. PQ, where represents the arc subtended by QMF at distance unity. Now the are whose radius is MT is represented by. MT; but this arc, before it was developed from the cone, formed a complete circle whose radius was NT, and therefore its circumference 2-NT; since then the circle has not altered its length by its development, we have * If the cone be supposed covered with a flexible sheet, and a band such as PQFK be imagined to be cut upon it, and then unwrapped from the cone and laid upon a plane, it is called the development of the band. @ MT=2-NT. Substituting this value of MT in the expression for the area of the band we have. area of zone PQFK=2. NT. PQ. Let the surface be conceived to be divided into an infinite. number of such elementary bands, and let the lengths of the corresponding elements of the curve AB be represented by 81, 82, 83, &c. and the corresponding values of NT by y, Y2,Y, &c. Then will the areas of the corresponding zones be represented by 2y,s,, 2,82, 277,8,, &c. and the area of the whole surface BAC by 2-y ̧§, +2*у,8, +2* y z 8, + . . . . or by 2 (y,8, +3,8, +7,8, + ....). But since G is the centre of gravity of the curved line AB, therefore AB. GH represents the moment of the weight of a uniform thread or wire of the form of that line about AD, being the weight of each unit in the length of the line: moreover, this moment equals the sum of the moments of the weights 8, 8, 8,4, &c. of the elements of the line. But 2-GH equals the length of the circular path described by G in its revolution about AD.. Therefore, &c. This proposition, like the last, is true not only in respect to a surface of revolution, but of any surface generated by a plane curve, which traverses perpendicularly another curve of any form whatever, and is always intersected by it in the same point. It is evident, indeed, that the same demonstration applies to both propositions. It must, however, be observed, that neither proposition applies unless the motion of the generating plane or curve be such, that no two of its consecutive positions intersect or cross one another. 41. The volume of any truncated prismatic or cylindrical body ABCD, of which one extremity CD is perpendicular to the sides of the prism, and the other AB inclined to them, is equal to that of an upright prism ABEF, having for its base the plane AB, and for its height the perpendicular height GN of the centre of gravity G of the plane DC, above the plane of AB. For let represent the inclination of the plane DC to AB: |