JoLY — Quaternmi Invariants of Linear Vector Functions, 8fc. 7 



8. On interchange of 7~oivs, six quaternion invariants are found ; these 

 are equivalent to one scalar, and three vector invariants, — The eSect of 

 the interchange of rows was partially considered in the 5th Article. 

 Closely connected Trith this is the effect of the interchange of ^i, ^jj 

 and 03 in the invariant q (^i, <^2, <^3)- In order to see the connexion, 

 it is only necessary to remark, that if 



tti = <Aia, a2 = 4*^^) scu^ a3 = ^sa, 



with similar meanings for ;Si, ^^2, ^3, and yi, y^, 73, 



o-i Pi yi 



q (<^i, <^2. S^s) ^a.py = as A 72 



"3 /?3 73 



For brevity, let /sC^i; ^2. ^s) lie denoted by 4. as in this scalar 

 part transposition of the functions is vrithont effect ; ^ then 



9. (^1) ^2, ^s) = 4 + 2 (</)i>723 - S^2'>731 + 03^12), 



i (^3> 02, 0l) = 4-2 (017723 - ^a^/si + 03^12) ; 



? (<^2J 03, 00 = 4 + 2 (0i->723 + 03%1 " 0:i''7l2), 



? (^Ij S^3, 02) =4-2 (01^723 + 02%1 - 03>Jl2) ; 



and 



? (<^3, 01, 02) = 4 + 2 (- 017723 + 02^31 + 03>7l2), 

 ? (S^2, 01, 03) = 4 - 2 (- 017723 + 007731 + 03'>?12)- 



The quaternions are here grouped in conjugate pairs, and the six 

 different values of determinants of the third order formed by the 

 same three rows in different orders are exhibited. 



9. Relations connecting vector invariajits. Two reducing formulcB. — 

 The six invariants lately considered are equivalent to one scalar and 

 three vector invariants. I propose now to consider some reductions 

 and relations concerning vector invariants. 



Ptetaining the suffix notation, let ei denote the spin-vector of 0i, 

 €2 that of 02, and Ci2 and cai those of 0i 02 and 02 0i, respectively. Let 

 01 satisfy the cubic 



01^ - nil 01- + m2 01 - nis = 0, 

 and let 0, satisfy 



02^ - mi'cjio^ + rn.'d)^ - mz = 0. 



1 This may be verified by expansion of the determinants, but it is otherwise 

 obvious. 



