50 Proeeedingn of the Royal Iris/} Academy. 



9. In the fourtli place, write 



r = if{pq)-mw) = G{Pi); (17) 



and we find an equation which represents a one-to-one correspondence 

 between lines pq and points r. 



10. Again, consider the mutual relations of three points p, q, 

 and r, which satisfy the equation 



^pf{qr) = 0. (18) 



If p = q = r, the equation 



Mixr) = 0, or SrP(rr) = 0, (19) 



represents the general cubic surface, and with this surface are asso- 

 ciated systems of linear complexes, 



Sj»(/(r)-/(r)) = 0, or ^pC{qr) = Q, (20) 



so that to each value of p corresponds a linear complex represented 

 by (19). 



This is quite analogous to the quadric surface and the correlated 

 linear complex 



^qfq = 0, Spfq - 8qfp = 0, (21) 



obtainable from a linear function/. 



Further, by suitable permutation of the quaternions in (18), we 

 may obtain an equation of the form 



(Ipqr) = 0, (22) 



which is combinatorial with respect to p, q, and r, where ^ is a 

 constant quaternion determined by the nature of the function/. This 

 equation (22) represents a determinate fixed plane which contains the 

 points^, q, and r. 



11. Similarly, for the trilinear function, various analogous results 

 may be^ deduced ; but there is one which deserves special mention. 

 The equation 



P = /(«, i, 9), (23) 



in which a and b are quaternion constants arbitrarily assignable, 

 represents the complete group of homographic transformations in space, or 



