A Treatise on the Theory of Screws

102 THU THEORY OF SCREWS. [110, 110. The Reciprocal Screw System. The integer which denotes the order of a screw system, and the integer which denotes the order of the reciprocal screw system, will, in all cases, have the number six for their sum (§ 72). Hence a screw system of the first order will have as its reciprocal a screw system of the fifth order. For a screw 6 to belong to a screw system of the fifth order, the necessary and sufficient condition is, that 6 be reciprocal to one given screw a. This condition is expressed in the usual form:— (p„ + Pø) cos 0 - daB sin 0 = 0, where 0 is the angle, and dae the perpendicular distance between the screws d and a. We can now show that every straight line in apace, when it receives an appropriate pitch, constitutes a screw of a given screw system of the fifth order. For the straight line and a being given, da(j and (J are determined, and hence the pitch pe can be determined by the linear equation just written. Consider next a point A, and the screw a. Every straight line through A, when furnished with the proper pitch, will be reciprocal to a. Since the number of lines through A is doubly infinite, it follows that a singly infinite number of screws of given pitch can be drawn through A, so as to be reciprocal to a. We shall now prove that all the screws of the same pitch which pass through A, and are reciprocal to a, lie in a plane. This we shall first show to be the case for all the screws of zero pitch*, and then we shall deduce the more general theorem. By a twist of small amplitude about a the point A is moved to an adja- cent point B. To effect this movement against a force at A which is per- pendicular to AB, no work will be required; hence every line through A, perpendicular to AB, may be regarded as a screw of zero pitch, reciprocal to a. We must now enunciate a principle which applies to a screw system of any order. We have already referred to it with respect to the cylindroid (§ 18). If all the screws of a screw system be modified by the addition of the same linear magnitude (positive or negative) to the pitch of every screw, then the collection of screws thus modified still form a screw system of the same order. The proof is obvious, for since the virtual co-efficient depends on the sum of the pitches, it follows that, if all the pitches of a system be * This theorem is due to Möbius, who has shown, that, it small rotations about six axes can neutralise, and if five of the axes be given, and a point on the sixth axis, then the sixth axis is limited to a plane. (“ Ueber die Zusammensetzung unendlich kleiner Drehungen,” Crelle’s Journal, t. xviii., pp. 189—212.) (Berlin, 1838.)