A Treatise on the Theory of Screws

123] FREEDOM OF THE SECOND ORDER. Ill are no real screws of zero pitch. If the pitch conic be a parabola, there is but one screw of zero pitch, and this must be one of the two screws which intersect at right angles. 123 Equilibrium of a Body with Freedom of the Second Order. We shall now consider more fully the conditions under which a body which has freedom of the second order is in equilibrium. The necessary and sufficient condition is, that the forces which act upon the body shall constitute a wrench upon a screw which is reciprocal to the cylindroid which defines the freedom of the body. It has been shown (§ 23), that the screws which are reciprocal to a cylin- droid exist in such profusion, that through every point in space a cone of the second order can be drawn, of which the entire superficies is made up of such screws. We shall now examine the distribution of pitch upon such a cone. The pitch of each reciprocal screw is equal in magnitude, and opposite in sign, to the pitches of the two screws of equal pitch, in which it intersects the cylindroid (§ 22). Now, the greatest and least pitches of the screws on the cylindroid are pa and pß (§ IS). For the quantity pa cos21 +pß sin21 is always intermediate between pa cos21 + pa sin21 and pß cos21 + Pß sin21- Hence it follows that the generators of the cone which meet the cylindroid in three real points must have pitches intermediate between pa and pp. It. is also to be observed that, as only one line can be drawn through the vertex of the cone to intersect any two given screws on the cylindroid, so only one screw of any given pitch can be found on the reciprocal cone. One screw can be found upon the reciprocal cone of every pitch from - oo to + oo . The line drawn through the vertex parallel to the nodal line is a generator of the cone to which infinite pitch must be assigned. Setting out from this line around the cone the pitch gradually decreases to zero, then becomes negative, and increases to negative infinity, when we reach the line from which we started. We may here notice that when a screw has infinite pitch, we may regard the infinity as either + or — indifferently. If we conceive distances marked upon each generator of the cone from the vertex, equal to the pitch of that generator, then the parallel to the nodal line drawn from the vertex forms an asymptote to the curve so traced upon the cone. It is manifest that we must admit the eylindroid to possess imaginary screws, whose pitch is nevertheless real. The reciprocal cone drawn from a point to a cylindroid, is decomposed into two planes, when the point lies upon the cylindroid. The first plane is normal to the generator passing through the point. Every line in this plane must, when it receives the proper pitch, be a reciprocal screw. The