62 Proceedings of the Royal Irish Academy. 



exactly come under the head of quaternions. The notion of a quaternion as 

 the quotient of two vectors is not immediately involved in the theory of 

 linear vector functions ; but it will probably be agreed that there is hardly 

 any part of Hamilton's wonderful " Elements of Quaternions " more instructive 

 and more useful than the chapters dealing with the functions of which we are 

 now speaking. 



After the lamented Professor Charles J. Joly had acquired that mastery 

 of quaternions which made him so appropriate an editor of Hamilton's book, 

 his attention was turned to the theory of screws, with results to which 

 reference has already been made several times in the present paper. In his 

 many writings, and in his correspondence with the present writer, he has 

 developed with abundant illustrations the intimate connexion between 

 quaternions and the theory of screws. Probably the most important and 

 instructive part of this work has been his exposition of the relations of the 

 screws of a system of the third order to a linear vector function. He has shown 

 how these theories are coextensive, and how every theorem with regard to the 

 screws of a S-system has as its counterpart a theorem with regard to a linear 

 vector function. The perfection of this analogy lies in the circumstance that 

 iu each case the theory is of the most general type. The theory of a system 

 of screws of the third order of the most general type corresponds to the 

 theory of a linear vector function of the most general type. The signilicance 

 of this circumstance will be appreciated if we remark that in the case of the 

 impulsive vector and the instantaneous vector already referred to, the linear 

 vector function which arises is not of the most general type. It is of that 

 special form which is known as self -conjugate. It seems therefore reason- 

 able to point out that the screw system of the third order is a geometrical 

 equivalent coextensive under all circumstances with the linear vector 

 function. 



In illustration of this statement, we may recall that nine data are required 

 for the complete specification of a 3-system ; for example, three data are 

 required for the centre of the pitch quadric, three more for the directions of 

 its axes, and three more for the pitches of its three principal screws. That 

 nine data are also required for the definition of a linear vector function is 

 also well known. Indeed the name of nonion has been proposed for this 

 function in consequence of the significance of this circumstance. 



Let (Xi, fix) ; (Xi, /u) ; (Xj, ft») be three pairs of vectors defining three screws 

 of a 3-sy8tem, and let a;,, Xi, x, be any three scalars: then /i, X will repre- 

 sent another screw of the same system if 



X =• x.X, + XjXi + -TiXi 



H = Xi^i + Xiflt + Xifl. 



