Jolt — Quaternion Invariants of Linear Vector Functions, 8fc. 5 



6. Quaternion invariant, linear with respect to each of three linear 

 vector functions, expressed as a quotient of determinants. — Having, per- 

 haps, sufficiently dTvelt on the manipulation, of these determinants "svith 

 non-commutative constituents, I shall now show that 



4>io- <;^i/5 <^iy 



(jina ^o/? ^27 

 e^stt ^s^ ^jy 



Safiy 



= 2 (s^i) ^2, <;^j) 



is a quaternion invariant of the three linear vector functions ^ in the 

 sense that it is independent of the vectors a, /?, and y. This may he 

 done hy expressing a, /?, and y in terms of any three vectors such as 

 i, j, and h, and using the methods indicated in Art. 3, or hy direct 

 expansion, which is to he preferred, as exhibiting more clearly the 

 dependence of the determinant on the linear vector functions. Thus, 



q ((^1, <^2, <^3) 'S'a/3y 



= 4^io- (</>2/5^3y - ^2y03/S) + <^ij8 {(^oy^^a. - ^2a<^3y) 



+ ^ly (^2a03^ - 02/3<}!)3a) 

 = ^3 (<^i, ^-2, ^z) Sa.l3y 



+ (piaS {cfio/Scjizy - (ji^y^il^) + <^i/5*S'((/)oy^3a - (^oacj^^y) 

 -t- <f)^yS ((/)2a03/3 - 02^03^) 



- ^3a^ (^a/^^r/ - <^jy<^i)8) - (f)o/3S (<^3y<^ia - S^sa^^iy) 

 - 0jy5(^3a0i/3 - cji^ftcfi^a) 



+ cji^aS {<f}il3cf)2y - <^iyi^iP) + i^i/3S{4>iycji2a - ^lac^oy) 



in which 



SVV'l) V2) 



Saf3y 



is a scalar invariant, noticed in Art. 22 of the Paper already 

 referred to. 



Xow, if ^2 is the conjugate of ^o, 



8 (<^2iS<^3y - 4>2y^zP) = 'S (<^3'«^2 - 4>^^z) (^.y = 2Sy,,/3y, 



if 7^23 is the spin vector or non-conjugate part of 4>3 4>2.'^ Hence, if 2 

 denotes summation for cyclical transposition of a, /?, and y, 



X4>iO.S (^2^<^3y - 4>zy4>zl^) = 2'2,(J3iaS-r].,_3l3y = 2<^i77o3 . ^SafSy. 

 * The vector functions <p3'<p2 and <p2'<p3 are conjugate, since 



SA<pi'(pilJ, = S<pz\([)2fJ. = SfJ.(p2'<pi\. 



