102 THE THEORY OF SCREWS. [HO, 



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 a + Pe) cos d ag sin = 0, 



where is the angle, and d ae the perpendicular distance between the screws 

 9 and a. 



We can now show that every straight line in space, 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, d a6 and are determined, 

 and hence the pitch p e 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*, arid 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 Mdbius, who has shown, that, if 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. (&quot; Ueber die Zusammensetzung unendlich kleiner Drehungen,&quot; Crelle s 

 Journal, t. xviii., pp. 189212.) (Berlin, 1838.) 



