8 Proceedings of the Royal Irish Academy. 



For an arbitrary vector p, 



= (<^i'<^2 - <^2'<^i) P + 2 rei<^2p + 2<^2' Fkip 



= 2 Ft/ziP + 2 (m/ Feip - r<^2eip) ; 



using the fundamental relation 



m/ Feip = Fei^2P + F<^2eiP + H ^^iP> 



and retaining the signification of •>72i- 



But p is arbitrary, and (^1^2 - 4>2'4>i) P = 2 Feizp, and conse- 

 quently, 



ei2 = - •'yiz + W'l'ei - ^2ei. 



Interchanging the functions <^i and <^2, the similar relation 



C21 = + ■>7l2 + Wilf2 - 01^2 



is found. 



If ^1 = ^2j as a particular case of these relations 



10. On cyclical transposition of a product of functions. — Adding the 

 two relations found in the last Article, 



C12 + €21 = m/ei + ^162 - ^2^1 - 4>i^i ; 



this formula will be found to be of importance in the reduction of 

 the number of vector invariants. By its means the spin- vector of any 

 function 6(^ may be expressed in terms of that of ^0, and of the 

 results of operation on the spin- vectors of 6 and <l>. More generally 

 by repeated application of the formula, the spin -vectors of any cyclical 

 group of functions such as (^"^0, 4>0(f>, ^<^^ (in which the symbols 6 and 

 (ji are cyclically transposed) may be expressed in terms of the spin- 

 vector of any one of the functions {cji'^O suppose) and in terms of the 

 results of operation on the spin- vectors of simpler functions. 



11. When a square enters into the product, the spin-vectors are 

 reducible. — Eeplacing ^2 by (^2<i>i in the first formulae of Art. 9 (which 

 may be written in the form 



r{^,^, - <^/</,/) = r(()f,/<^2 - «^2>i) + m, (c/,2) . r(c^i - </>/) - ^. r{4>, - «^/)) 



^(<^l«j!>2<Al - «^lW<^l') = F(</>/<^2<^l - <^lV<^l) 



4- «h (</.2<^o r{4>, - </>/) - </.2(^i r(<^i - <^i') 



is the result, m^ {^i4>\) denoting the m^ invariant of <^2^i. 



