1902.] 



Quaternions and Projective Geometry. 



177 



"Quaternions and Projective Geometry." By Charles J. Joly,. 

 F.T.C.D., Eoyal Astronomer of Ireland. Communicated by 

 Sir Robekt S. Ball, F.B.S. Eeceived November 27, — Bead 

 December 11, 1902. 



The object of this paper is to include projective geometry within 

 the scope of quaternions. The calculus, as established by Hamilton, 

 was solely adapted to the treatment of metrical relations, but when we 



the point represented by the quaternion, q, is the extremity of the 

 vector, Vq/Sq, drawn from an arbitrary origin, and the weight 

 attributed to the point is Sq ;* and this interpretation requires no 

 modification in the principles of the calculus. 



In this paper the theory of the linear quaternion function is 

 developed to a considerable extent, and this theory is of fundamental 

 importance, because the most general nomographic transformation in 

 space is expressible by means of a linear quaternion function. One 

 section of the paper is devoted to the consideration of the scalar 

 invariants of linear quaternion functions, and among these are included 

 the invariants of systems of quadrics which correspond to the particular 

 case in which the functions are self-conjugate. Moreover, in this 

 section and more fully in the section on covariance, it is pointed out 

 that the invariance in the case of the general functions is wider than 

 in the case of self-conjugate functions. In fact, in the special case, 

 the functions must remain self-conjugate after transformation. It is 

 shown that there are in all eight distinct types of covariance. 



The decomposition of linear transformations is also considered, and 

 much use is made of the square-root of a linear quaternion function. 

 A section is occupied with the determination of the linear trans- 

 formations which shall convert given figures into other given figures, 

 and with the conditions which in certain cases must be obeyed in 

 order that such a transformation may be possible. 



The general surface, the principle of reciprocity, generalised curva- 

 ture and geodesies, are dealt with in a subsequent section, and the 

 chief properties of an operator analogous to Hamilton's y are exhibited 

 in the section immediately following. 



Several sections are occupied with the theory of the bilinear quater- 



( Abstract.) 



* 'Trans. JJ. Irish Acad./ rol. 32, pp. 1—16. 



