110 



Proceedings of the Royal Irish Aeademy. 



From a single function (/>, two functions / of the second order are- 

 obtained, defined by 



/i V^Kjji = Vn {X4>ix + ^A/x), and /, VoXfjL = F'2<^A<^/x. 



These are the analogues of Hamilton's functions, denoted by x ^^^^ "A'? 

 respectively, and their conjugates are 



f\ KX/x = Fo (A<^> + cji'Xix), and fo KX/x = r2<^'A<^>. 



Functions of th.e third order depending on a single linear vector 

 function <^ of the linear, or ordinary kind, are 



f^Y^XfJiv = y^{4'^H'^ + X(jifjiv + Xfj.cftr), 

 f-^T^'^Xfxv = F^(<^X<^/xj/ + (pXfjicjiv + (pXtp/xv), 

 and /a V^X/jlv = F'ge^A^/xc^v. 



These are the invariants nii, nio, and m^ of <^ when but three units are- 

 involved, or the coefiScients in the symbolic cubic 



<^^ - niicfy + niof^ 



0. 



Following the notation used in a Paper on " Quaternion Invariants- 

 of Linear Vector Functions " (Proc. Poy. Irish Acad., 1896), the func- 

 tions of the type here considered may be expressed by the general 

 equation 



«1) «2) Ct3j • • • O-N 



/• 



o-i, 0.2, aa, 



a.V 



a-N 







^yo-\, (jiya^, . . . (^iVa^v 



in which the determinant' on the left-hand side, operated on by f, 

 consists of the same row of iV^ vectors repeated iV times; the determi- 

 nant on the right consists of rows of these vectors operated on by iV 

 ordinary linear functions ^i<^2 . . . 4>]sr. 



If iVis equal to the number of units involved, the functions (/) 

 degenerate into invariants. 



When only a single function <fi is involved, the axes of these 

 functions are Vjv (a product of iV^axes of ^), and the roots are sums 

 of: — (1) the JV corresponding roots, (2) products of these in pairs, 

 (3) products in threes, &c. 



^ A eonyention must be adopted in the expansion of these determinants. It may 

 be comprised in the rule : — Expand as if the constituents were scalar, but preserve 

 the order of the rows. 



