Ball — Conlributions to the Theorij of Screws. 2t 



Let 1', Q bo the two points in wliiuii any two screws « and /3 are 

 respectively intersected by their common perpendicular. If we imagine a 

 to be rotated in a right-handed direction about the vector PQ until it 

 becomes parallel to /3, the angle tlirough wliicli a has been turned is the 

 right-handed angle between /3 and the original position of a. But if we then 

 continue the rotation of a for another 180° in the same direction, it will, in 

 the absence of the indicating vectors, resume precisely the same position 

 with regard to /3, so that 180° + 9 has just as much claim as to be regarded 

 as the right-handed angle between the screws a and |3. If in the expression 

 already found for the virtual coefficient (2) we increase 9 by 180°, the 

 magnitude is unaltered though the sign is changed. 



Thus the virtual coefficient of two screws is definite in magnitude but 

 indefinite in sign. In this respect the virtual coefficient of two screws offers 

 a parallel to the cosine of the angle between two lines. 



The virtual coefficient of two vector-screws is definite in magnitude and 

 not indefinite in sign. This is, of course, parallel to the case of the cosine 

 of the angle between two vectors. 



The only indefiniteness in the right-handed angle between two screws 

 which do not intersect is an integral multiple of 180°. 



If two vector-screws are reciprocal, they will remain reciprocal if the 

 directions of either or both of the vectors are reversed. In speaking of 

 reciprocal vector-screws we may therefore omit the word 'vector'; for tlie 

 relation indicated is irrespective of the sense of direction on either screw. 



II. — On the Composition of T'wisis or Wrenches on Vector-Screws. 



Let a', j3', 7' be the amplitudes of the twists on three vector-screws 

 a, j3, 7 ; the relations between the three amplitudes and the three screws 

 being such that the body after the last twist is restored to the same place 

 which it occupied before the first. 



Let »)" be the intensity of a wrench on any fourth vector-screw »;. Then 

 the virtual moment of this wrench wliile the body receives the twist a' is 

 '2i]"a'wa.-^. Hence the total work done in the course of the tliree twists is 



2»)"a'ra„,, 1 2i("/3't3^, + 2ij"7'-53^,. (3) 



As the body is restored to its original position after the completion of the 

 three twists, the expression just written must be zero, whatever be the surow ij 

 or whatever be the magnitude >)" : we therefore have* 



a''^„n + /3'to^, + 7'3J„ = . . . (4) 



• "Treatise," p. 18. 



