10 Proceeding-^ of the Royal Irish Academy. 



Hence, for any rLumber of funetions, it is only necessary to con- 

 sider th.e cycles in widch. no function occurs twice. 



Tor three functions these cycles are to be derived from 



0-3^3, <}>3<f>i, 4'i4*2 '} 4*i4^z4^ ^lid oj6;d)i. 



13. SearcA for new vector invariants niay he limited to the considera- 

 tion of ijiin-vectors. — Tiiere is no difficulty in seeing that the vector 

 parts of any of the quaternions 



?(^^^, 1, 1), ^(<i2^, "in 1), ?(<^i, ^, ^3), &c., 



are linear and invariant functions of the spin -vectors of the five cycles 

 of the last article, and of the spin-vectors of 61, c^, and ^3. The 

 vectors involved in q{dto(i>z, (j>i, 1) are 



r(0/^.<^-^'^'d>i), «^F(^,^-^'^0, and <^2<^F(<^i-0/). 

 The first of these is, by Art. 9, expressible in terms of 



and results of operation. 



Hence, in searching for new vector invariants, it is legitimate to 

 investigate the spin- vectors alone of the functions formed by multi- 

 plying the given functions together. There is no need to investigate 

 separately the spin-vectors of products such as <^i'^2<^j. 



14. Reducing systems of quaternion invariants. — The "reducing 

 systems '"' of the previous Paper may be used in the more general 

 case of quaternion invariants. It is evident, from the constitution 

 of these functious, that 



xq (j/r,^V2! ^i, ^i) + yq {^i4>^2^ ^3, ^i) + zq («/^i«^2, ^3, tl^i) 



= 9 L^i (^«f>" T y<^ + z) 1/^2, ^3, ^i], 



in which x, y, and z are scalars, and ^ and the ij/ linear vector func- 

 tions. Xow, as <}> satisfies a cubic equation 



<^' — rni<f>- -r ??^2^ — t/u = 0, 



any rational algebraic function of may be reduced to the form 



f{(j)y = :r<f»- -^ y(f> -t- z, 

 and therefore 



q ("Ai/C^) "A2. ^z, ^i) 

 is reducible in terms of three simpler quaternions. 



