DYNAMICS OF A RIGID BQDY. 85 



pitch, will be reciprocal to a. Since the number oflines 

 through A is doubly infinite, it follows that a singly in- 

 finite 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 y 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 adjacent point B. To effect this movement 

 against a force at A which is perpendicular to AJB, no 

 work will be required; hence every line through ^4, per- 

 pendicular to A By may be regarded as a screw of zero 

 pitch, reciprocal to a. 



We must now enunciate a principle which applies to 

 a screw complex of any order. We have already re- 

 ferred to it with respect to the cylindroid (20). If all 

 the screws of a screw complex be modified by the ad- 

 dition of the same linear magnitude (positive or nega- 

 tive) to the pitch of every screw, then the collection of 

 screws thus modified still form a screw complex 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 complex be increased by a cer- 

 tain quantity, and all the pitches of the reciprocal com- 

 plex be diminished by the same quantity, then all the 

 first group of screws thus modified are reciprocal to all 

 the second group as modified. Hence, since a screw 



* This theorem is due to Mobius, 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. (Ueber die Zusam- 

 mensetzung unendlich kleiner Drehungen Crelle's Journal, t. xviii., pp. 189- 



