STATICS. CHAPTER I. CENTRE OF GRAVITY. LET dm represent an element, at any point x, y, z, of the mass of a body referred to any three co-ordinate axes, rectangular or oblique, and let x, y, z, denote the co-ordinates of the centre of gravity of the body; then the formulæ for finding the values of x, y, z, are the limits of the integrations being determined by the form of the body. If the body be bounded by a surface represented by a single algebraical equation in x, y, z, the evaluation of each of the will require the perform expressions fædm, fydm, fzdm, fàm, ance of the operation of integration on a single function of x, y, z, between appropriate limits; if, however, the body be bounded by discontinuous surfaces, the evaluation of each of these expressions will require the integration between proper limits of several functions of x, y, z, corresponding to the several discontinuous surfaces; the sum of the definite integrals of these functions being the required value of the expression. The idea of the centre of gravity of material bodies is due to Archimedes, by whom the centres of gravity of various areas W. S. 1 were investigated in his treatise, entitled Επιπέδων ἰσοῤῥοπικῶν ἢ κέντρα βαρῶν ἐπιπέδων. He likewise determined the centre of gravity of the parabolic conoid. Among the mathematical successors of Archimedes who have cultivated the science of the centre of gravity, may be mentioned Pappus1, Guido Ubaldi, Lucas Valerius, La-Faille*, Guldin', Wallis, Carré', Varignon, Clairaut9. SECT. 1. Symmetrical Area. Let x be the abscissa and y the ordinate of any point in the circumference of a plane area, symmetrical with respect to the axis of x; the axes of co-ordinates being either rectangular or oblique. Then the centre of gravity of any portion of this area, intercepted between any assigned pair of double ordinates, will lie in the axis of x, and its distance from the origin will be given by the formula Jxy xy dx where the integrations are to be performed between limits depending upon the positions of the intercepting ordinates. The value of is sometimes more readily obtained by polar co-ordinates, when the formula will be where r denotes the distance of any point within the area from 1 Mathemat. Collect., lib. 8, published for the first time in 1588. 2 In duos Archimedis Equiponderantium libros Paraphrasis, 1588. 3 De Centro Gravitatis Solidorum, 1604. De Centro Gravitatis partium Circuli et Ellipsis Theoremata, 1632. 5 Centrobaryca, 1635. 6 Opera, tom. 1. cap. 4 et 5, 1670. 7 Mésure des Surfaces, 1700. 8 Mém. de l'Acad. des Sciences de Paris, 1714. 9 Mém. de l'Acad. des Sciences de Paris, 1731, p. 159. the origin, and the inclination of r to the axis of a. The nature of the limits in the double integrations will depend upon the form of the area in each particular case. Supposing the area to consist of several portions, the boundaries of which are defined by distinct equations, the above formulæ must be replaced by where represents the summation of the integrations performed in regard to the several portions of the area. (1) To find the centre of gravity of the area of any portion BAC (fig. 1) of a parabola cut off by any chord BC. Let Py be the tangent to the parabola, which is parallel to the chord CB, P being the point of contact; from P draw Px parallel to the axis of the parabola. Then, Px and Py being taken as the axes of x and y, the equation to the curve will be y2 = 4mx, m being the distance of the point P from the focus. Archimedes, 'ETIπédwν iσоррожIKOV, Lib. 11. Prop. 8; Guldin, Centrobaryca, Lib. I. cap. 9, p. 121. (2) To find the centre of gravity of the area of the Cissoid of Diocles, EAE', (fig. 2). (3) To find the centre of gravity of the sector ABC (fig. 3 of a circle, of which C is the centre. From C draw the straight line CEx bisecting the sectoria area; and draw Cy at right angles to Cx. Let CE= a, and A Ca=a; then, Cx, Cy, being the axes of x and y, Now the equations to the straight line CA, and to the circl of which AEB is an arc, are respectively y=x tan α, y2 = a2 - x2; |