18 Proceedings of the Royal Irish Acadenvj. 



has now to be made. By a vector-screw, which is here defined for the first 

 time, I mean a screw in the original sense of the word,* on the axis of which 

 a unit vector is laid. Thus a vector-screw differs from a screw in that the 

 former possesses an indication as to which of the two directions along the 

 axis is to be regarded as positive, while there is no such indication of the 

 positive direction in the latter. It must not be supposed that the vector by 

 which a particular direction is indicated as positive on a vector-screw stands 

 in any relation to the pitch of the screw. The sign of the pitch may be 

 positive or negative, but is quite irrespective of the sense of direction 

 imparted to the vector-screw when it carries a unit vector. Of 

 course the pitch of a vector-screw may be zero, and two quite distinct 

 vector-screws of zero or any other pitch may lie on the same axis with 

 their unit vectors in opposite directions. 



Five data are in general sufficient to determine a screw ; but it would not 

 be quite correct to say that five data suffice to determine a vector-scrcv:. We 

 have just seen that every screw will be the seat of two distinct vector-screws 

 according to the direction of the vector. Thus five data, though insufficient to 

 specify a vector-screw complete, will yet show that the vector-screw must be 

 one of a definite pair of vector-screws of the same pitch and on the same axis.t 



The distinction between a right-handed rotation? about a vector and a 

 left-handed rotation about a vector is perhaps most becomingly based on the 

 relations of the diurnal rotation of the Earth to the North and South Poles 

 of the Elarth. We accordingly distinguish the right-handed rotation from 

 the left-handed rotation as follows : — 



The Diurnal rotation of the Earth is said to be riyht-handcd about a 

 vector from its centre to the North Pole, and left-handed about a vector from 

 its centre to the South Pole. 



It is to be understood that when a body is said to have had a right- 

 handed rotation about a vector it is implied not only — 



(1) That the body has received a rotation about that vector as an axis ; 

 but also 



• "Treatise," p. 7. 



t It will be seen a little later that when a screw is represented by the rettor coordinates /i, A 

 the geometrical fonn indicated is really a vector-screw and not merely a screw. Thus the 

 quaternion method of representing Dynames is free from the present ambiguity. 



X See Hamilton, ''Lectures on Quaternions," art. 68; also Hamilton, "Elements of Quater- 

 ninns," 2nd ed., edited by Charles Jasper Joly (1899), toI. i., p. 21.5, foot-note. It is this edition 

 of the great work which will be referred to throughout this paper whenerer Hamilton's " Elements 

 of Quaternions" is quoted. On the subject of the convention respecting the direction of a right- 

 handed rotation about a vector, reference may be made to " A Manual of Quaternions " by Phnrles 

 Jasper Joly, 1905, p. 7. This work will be quoted briefly as Joly's "Manual" in the frequent 

 references made to it in the prCMrnt pap«r. 



