JoLY — Quaternion Invariants of Linear Vector Functions, 8fc. 9 



and therefore as p is arbitrary, 



The spin-vector of <^i'<^2<^i is consequently reduced to a result of 

 operation on the spin-vector of ^2. 



To see the full bearing of this, observe that by a formula lately 

 ■written, 



r(<^i^2<^i - ^i'<^2'<^/) = 2w3<^f 'ez + 2»h (</)30i) . ei - 2cf)^4>i^j,' 



Thus the spin-vector of 4>i4>'i4>i is likewise reduced to the results of 

 operation on the spin-vectors of simpler functions, and therefore, by 

 the last Article, the spin-vectors of (f'l'ffii and (f)24>i^ are similarly 

 reducible. Generally, therefore, having formed from any number of 

 functions <}> a function $ = <^i<^2^3^i<Ai &c., the spin-vectors of all 

 the functions formed by cyclically transposing the ^ in $ are, by 

 the last Article, linearly and invariantally expressible in terms of 

 the spin-vector of $, and in terms of the results of operation on 

 the spin-vectors of simpler functions; and, by the present Article, 

 if any one of the <^ is consecutively repeated in $, the spin-vectors 

 of the functions of the group are all expressible in terms of the 

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



12. This is also the case when the same function occurs twice in the 

 product. — For two functions, ^i and ^3, the vector invariants are the 

 results of operation on the two spin-vectors ci and e.^, and on one of the 

 three vectors 612, eji, and 7712- In this case the cyclical group consists 

 of the functions <^i(^2 and 4>2.4>i- The group ^i<^2<^i<^2 gives a reducible 

 invariant. Before proceeding to the consideration of three functions, 

 another general formula of reduction will be given. 



If and \l/ are any linear vector functions, the vector invariants of 

 the cycle ^i^^i^^ are reducible. A function of the cycle is xpc^iOt^i and 



by a formula of Art. 9 or 11, the functions ij/cjii and 6(^1 replacing 

 ■^1 and ^2- 



Now, as in the last article, 



so V{4>,'f. eel,, - <f,,'0' . iA</)0 = m, (<^o . <^-i r(f ^ - ^v)> 



and the theorem just stated is proved. 



