A Treatise on the Theory of Screws

391] THE THEORY OF PERMANENT SCREWS. 429 If the restraints are such that the number of degrees of freedom is less than four, then an infinite number of rigid bodies can be designed, such that the impulsive screws and their corresponding instantaneous screws shall be represented by a given chiastic homography. If n exceed four then it will not in general be possible to design a rigid body such that its corre- sponding impulsive screws and instantaneous screws shall agree with a given chiastic homography. If, however, n = 4 then it is always possible to design one but only one rigid body so that its pairs of corresponding impulsive screws and instantaneous screws shall be represented by a given chiastic homography. Returning to the three-system we may remark that, having settled the inertia conic in the plane representation we are not at liberty to choose three arbitrary points as representing the three permanent screws. For if these three points were to be chosen quite arbitrarily, then six relations among the co-ordinates of the rigid body would be given, and the conic of inertia would require five more conditions. Hence the co-ordinates of the rigid body would have in general to satisfy eleven conditions which, of course, is not generally possible, as there are only nine such co-ordinates. It is therefore plain that when the conic of inertia has been chosen at least two other conditions must necessarily be fulfilled by the three points which are to represent the permanent screw. This fact is not brought out by the method of § 389 in which, having chosen the three permanent screws arbitrarily, we have then written down the general equation of a conic as the inertia conic. This conic should certainly fulfil at least two con- ditions which the equations as there given do not indicate. We therefore calculate directly the expression for the kinetic energy of a body in the position 0/, 0f 0J twisting about a screw with twist velocities Öi, 0„, 03 when the screws of reference are the three principal screws of the three-system with pitches a, b, c, and when #0, y0, If, If If pf pf pj are the nine co-ordinates (§ 324) of the rigid body relative to these axes. It is easily shown that we have for the kinetic energy the mass M of the body multiplied into the following expression where squares and higher powers of 0f0f 03' are omitted:— I (a3#,2 + b20J + c20J + pJ0J + pJ0J + pJ0J) + 0203 (c - b)æ0+ 0S0! (a -c)y0 + 0t02 (b — a) 20 - 0203f - ÖÄU - 0j2lj "+ a (ØJ + Øf) æ0 + (ØJ — Øf) If + 0203 (pJ - pJ + ac - ab) _+ (Ij — Cøj) — ØJØ‘2 (42 +