A Treatise on the Theory of Screws

THE THEORY OF SCREWS. whence 404 [363- but from the physical property of the kinetic energy already cited, it appears that this kind of displacement cannot change the kinetic energy, whence ödT + 0 01 dø,' - + øn døn'~°- 364. The Converse Theorem. Let us take the general case where the co-ordinates are and ... xn. Suppose that x2,... xn are all zero, then x, is the velocity of the mass-chain. We shall also take x2,... xn to be zero, so that we only consider the position of the inass-chain defined by x,. Think now of the two positions for which x, — 0 and x, = x,' respectively. Whatever be the character of the constraints it must be possible for the mass-chain to pass from the position «i = 0 to the position x, = x^ by a twist about a screw-chain. The magnitude x, is thus correlated to the position of the mass-chain on a screw-chain about which it twists. If the co-ordinates are of such a kind that the identical equation which T must necessarily satisfy has the form . dT . dT n X1dxr"+Xndx,;~Q’ then for the particular displacement corresponding to the first co-ordinate, x2, ... xn are all zero, and ^=o; dx1 and as T must involve x, in the second degree, we have T = Hx,* where H is independent of x,'. Let 0, be the twist velocity about the screw-chain corresponding to the first co-ordinate, then, of course, A being a constant, T = A0,\ A0,2=Hx^, JÄ0, = ^Hx„ and by integration and adjustment of units and origins 0i -x,. We thus see that while the displacement corresponding to the first co-ordinate must always be a twist about a screw-chain, whatever be the actual nature of the metric element chosen for the co-ordinate, yet that when the identical equation assumes the form 0dI dT -A Ø1do'--+Øndø,'-^