Page:The New International Encyclopædia 1st ed. v. 04.djvu/497

* CENTRAKCHID^. 425 CENTRE OF GRAVITY. and food-fish ot foiisideiablo importance. They build nests by eleaninj; away an area of gravel or sand, and wateh tlieni until the young are able to swim. See B.SS; Brk.vm ; C.lico-15ass ; t'KArriE; Rock-Bass; Sixfisii, etc. CENTRE (Fr. centre, Lat. centrum, Gk. kIi'. Tpnv, kentron, centre, from ncirciv, kentein, to prick). A term variously used in mathematics. (For centre of a cunt, see Cirves. ) Centre of a pencil, the verte. of a pencil of lines in a plane. Similarly the vertex of a sheaf of lines is called the centre. Centre of involution, a point in a range of points in involution such that the product of its distances from any two corre- sponding ])oints is constant. Centre of curva- ture of a plane curve at any point, the centre of the osculating circle (q.v.). Similarly the cen- tre of curvature of a surface is the centre of the osculating sphere. Centre of perspective, homologii. collincation, the point in which all lines joining pairs of corresp<inding points in perspective figures are concurrent. In case the figures are congruent the centre of perspective is called the centre of symmetrj-. (For centre of similaritti, or similitu'de, see Similarity.) Cen- tre of mean position or mean centre of points o»i « line, a point from which the algebraic sum of the distances to the given points is zero. The mathematical notion of centre of mean position corresponds to the mechanical idea of centre of mass and centre of gravity: e.g. the mean centre of the vertices of a triangle, i.e. the centroid, corresponds to the centre of mass of a homo- geneous Irianuuhir plate. CENTRE OF BUOYANCY. See BuoTANcy. CENTRE OF GRAVITY. Owing to gravita- tion (([.v. I, all bodies on the surface of the earth are being acted on by forces drawing them toward the centre of the earth. These forces are all sensibly parallel, owing to the large size of the earth compared with that of most natural objects, and produce what is ordinarily called 'weight.' The weight of any particle of matter whose mass is »i is therefore equal to mg, where <7 is the acceleration which the body would have toward the earth if allowed to fall freely. The value of <7 is jiroved by expeiiment to be the same for all kinds and quantities of matter, but to vary from point to point on the earth's sur- face. Any large body may lie considered as made up of parts, all being acted upon by paral- lel forces, and the resultant of these forces will be the weight of the body, and will be a force whose value, position, and direction are given by the ordinary laws for compounding jiaratlel forces. (See ^Ieciiamcs.) The point in the body (or in space connected with tlie bod}-) through which this re- sultant always ])asses, however the body is turned or placed, is called its 'centre of gravity.' Thus, if nij and ni^ are the masses of two small bodies, ■ " '' which may be called X 'particles,' kept at a I distance h apart, their cx'ntre of gravity may be calculated as follows: Through any point <) in the vertical plane in- cluding the particles, draw two straight lines. OX and OV, at right angles and parallel re- spectively to the vertical lines representing the weights of VI, and ni,. Let x, and x, be the ]>cipendicular distances of m, and nu from OY ; then, by the law of parallel forces, the resultant of the two forces 111,(7 and m^ is a parallel force (m,-f 7(12)3 at a distance x from OY, where m,rix, +m29 Xow, if the two bodies be moved in space, still keeping their distance h apart, OX and OY moving with them, they ■ " can be so placed that OX is now- vertical, as in the diagram. Call the distances of in, and }«. from OX 17, and j/j. The resultant now is a force (m, -|- >ii:)<7 parallel to OX. and at a distance y such that m,9 — X,- 5. "'i.7.'/i Hence V- (nil + vu)g "*! "I" '"2 It is evident from geometry that in both cases ihe resultant passes through a point on the line joining the two particles wdiose distance from the one of mass m is -h (Hj -f- m. This may be shown by choosing to coincide with the particle whose mass is hi. In that case Xi = 0, 17, = 0, and therefore ' m, -{- in. "^2 . y =; -Vi And so, by similar triangles, these condi- tions are satisfied by a point C on the line 00' such that 0C= •00 1 + "'2 -X This point is, there- ""i 1 fore, the centre of V gravity, being inde- pendent of the direction of the line 00". It is evident, further, from these equations for x and !/, that the centre of gravity coincides with the 'centre of inertia' (q.v.). The centre of gravity of any number of particles may be found in a perfectly similar way. For a imiform straight wire or rod the centre of gravity is its middle point; for a triangular plane figure it i:i the intersection of the three bisectors of the sides drawn from the opposite vertices: for a homogeneous pyramid it is the point of inter- section of the lines drawn from each vertex to the centre nf gravity of the opposite face. If a soliil body is suspended by a string fas- tened to it. or if it is balanced on a point, the line of action of this upwanl supporting force must pass through the centre of gravity, if the body is at rest. This gives, therefove, a direct method of determining the position of the centre of gravity of a .solid by experiment: Suspend it