CENTRE OF MAGNITUDE. 25 proceed as follows:-Divide the circumference exactly, by plane geometry, into such a number of equal arcs as may be required, in order to give sufficient precision to the approximative part of the process. Let the number of equal arcs in that preliminary division be called n. Divide one of them, by means of Rule V., into the required number of equal parts; n times one of those parts will be one of the required equal arcs into which the whole circumference is to be divided. Rules I., III., and V., are applicable to arcs of other curves besides the circle, provided the changes of curvature in such arcs are small and gradual. 39. To Measure the Length of any Curve.--Divide it into short arcs, and measure each of them by Rule I. of Article 38, page 23. SECTION 4.-GEOMETRICAL CENTRES AND MOMENTS. 40. Centre of Magnitude-General Principles.-By the magnitude of a figure is to be understood its length, area, or volume, according as it is a line, a surface, or a solid. The centre of magnitude of a figure is a point such that, if the figure be divided in any way into equal parts, the distance of the centre of magnitude of the whole figure from any given plane is the mean of the distances of the centres of magnitude of the several equal parts from that plane. The geometrical moment of any figure relatively to a given plane is the product of its magnitude into the perpendicular distance of its centre from that plane. I. Symmetrical figure.-If a plane divides a figure into two symmetrical halves, the centre of magnitude of the figure is in that plane; if the figure is symmetrically divided in the like manner by two planes, the centre of magnitude is in the line where those planes cut each other; if the figure is symmetrically divided by three planes, the centre of magnitude is their point of intersection; and if a figure has a centre of figure (for example, a circle, a sphere, the centre of the arc to that summit will bisect the arc. (2.) To mark the sixth part of the circumference of a circle. Lay off a chord equal to the radius. (3.) To mark the tenth part of the circumference of a circle. In fig. 12, draw the straight line A B=the radius of the circle; and perpendicular to A B, draw B C A B. Join A C, and from it cut off C D C B. AD will be the chord of one-tenth part of the circumference of the A circle. (4.) For the fifteenth part, take the difference between one-sixth and one-tenth. It may be added that Gauss discovered a method of dividing the circumference of a circle by geometry exactly, when the number of equal parts is any prime number that is equal to 1 + a power of 2; such as 1+2=17; 1+2*=257, &c.; but the method is too laborious for use in designing mechanism. Fig. 12. B an ellipse, an ellipsoid, a parallelogram, &c.), that point is its centre of magnitude. II. Compound figure.—To find the perpendicular distance from a given plane of the centre of a compound figure made up of parts whose centres are known. Multiply the magnitude of each part by the perpendicular distance of its centre from the given plane; distinguish the products (or geometrical moments) into positive or negative, according as the centres of the parts lie to one side or to the other of the plane; add together, separately, the positive moments and the negative moments: take the difference of the two sums, and call it positive or negative according as the positive or negative sum is the greater; this is the resultant moment of the compound figure relatively to the given plane; and its being positive or negative shews at which side of the plane the required centres lies. Divide the resultant moment by the magnitude of the compound figure; the quotient will be the distance required. The centre of a figure in three dimensions is determined by finding its distances from three planes that are not parallel to each other. The best position for those planes is perpendicular to each other; for example, one horizontal, and the other two cutting each other at right angles in a vertical line. To determine the centre of a plane figure, its distances from two planes perpendicular to the plane of the figure are sufficient. 41. Centre of a Plane Area. To find, approximately, the centre of any plane area. Rule A.-Let the plane area be that represented in fig. 7 (of Article 34, page 17). Draw an axis, A X, in a convenient position, divide it into equal intervals, measure breadths at the ends and at the points of division, and calculate the area, as in Article 34. Then multiply each breadth by its distance from one end of the axis (as A); consider the products as if they were the breadths of a new figure, and proceed by the rules of Article 34 to calculate the area of that new figure. The result of the operation will be the geometrical moment of the original figure relatively to a plane perpendicular to A X at the point A. Divide the moment by the area of the original figure; the quotient will be the distance of the centre required from the plane perpendicular to A X at A. Draw a second axis intersecting A X (the most convenient position being in general perpendicular to A X), and by a similar process find the distance of the centre from a plane perpendicular to the second axis at one of its ends; the centre will then be completely determined. Rule B.-If convenient, the distance of the required centre from a plane cutting an axis at one of the intermediate points of divi CENTRE OF MAGNITUDE OF A CURVED LINE. 27 sion, instead of at one of its ends, may be computed as follows:Take separately the moments of the two parts into which that plane divides the figure; the required centre will lie in the part which has the greater moment. Subtract the less moment from the greater; the remainder will be the resultant moment of the whole figure, which being divided by the whole area, the quotient will be the distance of the required centre from the plane of division. Remark. When the resultant moment is = 0, the centre is in the plane of division. Rule C.-To find the perpendicular distance of the centre from the axis A X. Multiply each breadth by the distance of the middle point of that breadth from the axis, and by the proper Simpson's Multiplier," Article 34, page 18; distinguish the products into right-handed and left-handed, according as the middle points of the breadths lie to the right or left of the axis; take separately the sum of the right-handed products and the sum of the left-handed products; the required centre will lie to that side of the axis for which the sum is the greater; subtract the less sum from the greater, and multiply the remainder by of the common interval if Simpson's first rule is used, or by of the common interval if Simpson's second rule is used; the product will be the resultant moment relatively to the axis A X, which being divided by the area, the quotient will be the required distance of the centre from that axis.* 42. Centre of a Volume.-To find the perpendicular distance of the centre of magnitude of any solid figure from a plane perpendicular to a given axis at a given point, proceed as in Rule A of the preceding Article to find the moment relatively to the plane, substituting sectional areas for breadths; then divide the moment by the volume (as found by Article 37); the quotient will be the required distance. To determine the centre completely, find its distances from three planes, no two of which are parallel. In general it is best that those planes should be perpendicular to each other. A D --E C B Fig. 13. 43. Centre of Magnitude of a Curved Line.-Rule A.-To find approximately the centre of magnitude of a very flat curved line.In fig. 13, let ADB be the arc. Draw the straight chord A B, which bisect in C; draw CD (the deflection of the arc) perpendicular to A B; from D lay off DE centre required. CD; E will be very nearly the * The rules of this Article are expressed in symbols, as follows:-Let x and y be the perpendicular distances of any point in the plane area from two This process is exact for a cycloidal arc whose chord, A B, is parallel to the base of the cycloid. For other curves it is approximate. For example, in the case of a circular arc, it gives DE too small; the error, for an arc subtending 60°, being about of the deflection, and its proportion to the deflection varying nearly as the square of the angular extent of the arc. Rule B.-When the curved line is not very flat, divide it into very flat arcs; find their several centres of magnitude by Rule A, and measure their lengths; then treat the whole curve as a compound figure, agreeably to Rule II. of Article 40, page 26. 44. Special Figures.-I. Triangle (fig. 14).-From any two of the angles draw straight lines to the middle points of the opposite sides; these lines will cut each other in the centre required;-or otherwise, from any one of the angles draw a straight line to the middle of the opposite side, and cut off one-third part from that line commencing at the side. Fig. 14. II. Quadrilateral (fig. 15).-Draw the two diagonals A C and BD, cutting each other in E. If the quadrilateral is a parallelogram, E will divide each diagonal into two equal parts, and will itself be the centre. If not, one or both of the diagonals will be divided into unequal parts by the point E. Let BD be a diagonal that is unequally divided. From D lay off D F in that diagonal = BE. Then the centre of the triangle F A C, found as in the preceding rule, will be the centre required. D Fig. 15. III. Plane polygon.-Divide it into triBangles; find their centres, and measure their areas; then treat the polygon as a compound figure made up of the triangles, by Rule II. of Article 40, page 26. IV. Prism or cylinder with plane parallel ends. Find the centres of the ends; a straight line joining them will be the axis of the prism or cylinder, and the middle point of that line will be the centre required. planes perpendicular to the area and to each other, and x, and y, the perpendicular distances of the centre of magnitude of the area from the same planes; then - [ [ x d x d y; y1 = See Article 29, page 16. fydxdy SPECIAL FIGURES. 29 V. Tetrahedron, or triangular pyramid (fig. 16). Bisect any two opposite edges, as AD and BC, in E and F; join E F, and bisect it in G; this point will be the centre required. VI. Any pyramid or cone with a plane base. Find the centre of the base, from which draw a straight line to the summit; this will be the axis of the pyramid or cone. From the axis cut off one-fourth of its length, beginning at the base; this will give the centre required. D H G Fig. 16. B VII. Any polyhedron or plane-faced solid. Divide it into pyramids; find their centres and measure their volumes; then treat the whole solid as a compound figure by Rule II. of Article 22. .D E B VIII. Circular arc.-In fig. 17, let A B be the arc, and C the the centre of the circle of which it is part. Bisect the arc in D, and join C D and A B. Multiply the radius C D by the chord A B, and divide by the length of the arc A D B; lay off the quotient CE upon CD; E will be the centre of magnitude of the arc. IX. Circular sector, CA D B, fig. 17.Find CE as in the preceding rule, and make C F = C E; F will be the centre required. F Fig. 17. X. Sector of a flat ring.-Let be the external and r' the internal radius of the ring. Draw a circular arc of the same angular extent with the sector, and of the radius and find its centre of magnitude by Rule VIII. 2 p3 r'3 3 r2 |