96 Proceedings of the Royal Irish Academy. 



It is proved then that when 



qKq = JTq . q = sccdar, and qlq = Iq . q = scalar, 

 the function q must he of the type 



? = !Zu) + $'(3)> 01' S = U) + ^(2); 



that is, q must be either even or odd in the units. 



All the conditions to be satisfied are not yet exhausted ; there- 

 remain 



$'ii)5'(3) = $'(3)S'(i), and ^-^j^ - q^^ = scalar, 



for an odd function ; or else 



2'(o)2'i2; = 5'(2)$'(o)> and «7(o; - £(2)2 = scalar, 



for an even function satisfying the general conditions of this article. 



As an example of a quartic satisfying all the conditions, the 

 function 



q = aiojJz + «315644«5«6 



does not appear to be resolvable into linear factors. Hence it would 

 seem that the conditions of this article do not reqxiire a function to be 

 thus resolvable. (Cf. the first paragraph of this article.) 



28. Much of the investigation in recent articles will be useful in 

 the consideration of the functions produced on operating by £ ( ) q~^ 

 on a vector. 



Let p and cr be any line vectors, and let 



P = qpq~^, and % = qa-q'^ ; 

 then P2 = qpq~^ • q<^q~^ = 2'po"2'""S and 2P = qo'pq"'^. 



Adding these j)roducts, 



PS + :§P = q {pa- + a-p) q'^ = q . q~^ (pa- + ap) = pa- + crp, 



because pa- + o-p is a scalar, and therefore commutative with q or q~^. 

 Thus, Pl§ + SP is always a scalar, when P and ^ have been generated 

 from line vectors. In particular, P^ = p^, and %~ = o-^. 



Also, as special cases of these general results, let Ii, I2 . , . !„ be 

 the functions generated from the unit vectors ?i, iz, . . . 4> and it is 

 evident that 



I^^ = L^ = &c.^I„^ = -l, and that IJ. + LI^ = &c, = 0, 



or these new functions obey the laws of the unit vectors. 



