A Treatise on the Theory of Screws

210] PLANE REPRESENTATION OF THE THIRD ORDER. 213 In a certain sense therefore a must intersect each of these four screws, and accordingly the cubic has to pass through the four points. To prove that a is a double point we write for brevity p=e^+ø^+e.^ Q■=pi0i + pJH + p,,0p L =p1al2 + p2a2 + p3a32, R = P&& + p2a.202 + p3a303, S = a101 + a.,0., + a303, H = a,2 + a22 + a33, and the equation is P (2HR —LS) —SQH = 0. Differentiating with respect to 0lt 02, 03 respectively and equating the results to zero we have 0 = 20x \2RH -LS- aHS] + a2 [2aPH -LP- HQ], 0 = 20, [2RH -LS- bHS] + «2 [2bPH-LP - HQ], 0 = 203 [2RH -LS- cHS] + a3 [2cPH-LP - HQ]. These are satisfied by 0, = ax, 0, = a,, 03 = a3 which proves that a is a double point. The cubic equation is satisfied by the conditions 0 = «101 + %202 + a303, 0 =Pi<*i0i +p-2a20-2 +p3a303. This might have been expected because these equations mean that a and 0 are both reciprocal and rectangular, in which case they must intersect. Thus we obtain the following result: If a1; a2, a3 are the co-ordinates of a screw a in the plane representation, then the co-ordinates of the screw which, together with a, constitute the principal screws of a cylindroid of the system are respectively p3-Pz Pi~p3 P2-P1 ot2 ’ a, ’ a3 ' The following theorem may also be noted. Among the screws of a three- system which intersect one screw of that system there will generally be two screws of any given pitch. For the cubic which indicates by its points the screws that intersect a will cut any pitch conic in general in six points. Four of these are of course the four imaginary points referred to already. The two remaining inter- sections indicate the two screws of the pitch appropriate to the conic which intersect a. The cubic P(2HR —LS) — SQH = 0,