A Treatise on the Theory of Screws

___ ___ ____________ _________ ___ ______________________ CHAPTER XV. THE PLANE BEPKESENTATION OF FREEDOM OF THE THIRD ORDER*. 198. A Fundamental Consideration. Let x, y, z denote the Cartesian co-ordinates of a point in the body referred to axes fixed in space. When the body moves into an adjacent position these co-ordinates become, respectively, x +8«, y + Sy, 2 + 82, and we have, by a well-known consequence of the rigidity of the body, Sx = a + gz — hy, Sy — b + hx —fz, Sz = c + fy~ gx, where a, b, c, f g, h may be regarded as expressing the six generalized co-ordinates of the twist which the body has received. If the body has only three degrees of freedom, its position must be capable of specification by three independent co-ordinates, which we shall call 0.2, 0s. The six quantities, a, b, c, f, g, h, must each be a function of 0lt 02, 03, so that when the latter are given the former are determined. As all the movements are infinitely small, it is evident that these equations must in general be linear, and of the type ct — 4" in which Alt A2) are constants depending on the character of the constraints. We should similarly have b = BA + AA + AA, and so on for all the others. It is a well-known theorem that the new position of the body defined by 0lt 02, 03 may be obtained by a twist about a screw of which the axis is defined by the equations a + gz - hy _ b + hx -fz _ c +/y - gx ~ ~7 g h Trans. Hoy. Trish Acad., Vol. xxix. p. 247 (1888).