A Treatise on the Theory of Screws

375] THE THEORY OF PERMANENT SCREWS. 411 This is obviously true unless it were possible for the determinant (PT PT PT di)2’ dÖtdöi’ ’ dØ,dØn d2T PT d01dÖ2 ’ d0.2dén d2T d2T dØ,dØn døn2 to become zero. Remembering that T is a homogeneous function of the quantities 0„... 0n in the second degree, the evanescence of the determinant just written would indicate that T admitted of expression by means of n — 1 square terms, such as i -Li + IP ... + L\ This vanishes if Zi = 0; L2 = 0, &c.; Z„_x = 0; each of these is a linear equation in 0„... 0n, and consequently a real system of values for ... 0n must satisfy these equations, and render T zero. It would thus appear that a real motion of the mass-chain would have to be compatible with a state of zero kinetic energy. This is, of course, im- possible ; it therefore follows that the determinant must not vanish, and consequently we have the following theorem If the screw-chains of reference be co-reciprocal, then the necessary and the sufficient conditions for 0 to be a permanent screw are that its co-ordinates 0„02,... 0n shall satisfy the equations dT = = 0 d0' døn' There are n of these equations, but they are not independent. The emanant identity shows that if n -1 of them be satisfied, the co-ordinates so found must, in general, satisfy the last equation also. 375. Conditions of a permanent Screw-chain. As the quantities 0f...0f are small, we may generally expand T in powers, as follows:— t 0i ± i -r • • • *'n n + 0ffTa + ... + 201'0.'T12 + ... . The equation dT = 0 d0,' therefore becomes '1\ + 20,'Tn + 20.'TVi+ ... = 0,