394 HITCHCOCK. 



must be the coefficient m 3 for the example of the last article. In fact 

 we then have, remembering Is= 2, 



G(i 3 j) = 4A s sB s s -3A*sB 3 s-12As(A 2 )sBs(B*)s+3A s (A>) s B s (B>) s 

 +QAs(A>)sB s (B%+8(A*) s (B 3 )s-e>(A*) s (B*) s 



which is the same as (131). 



Developments like (145) are the same in form no matter what the 

 order of the matrices involved. They may in general be simplified 

 by the use of the Hamilton-Cayley equations for A and B. 



G(ij z ) may be obtained from G(i 3 j) by interchange of subscripts. 

 For the middle term of (144) we have 



G(»V) = i 2 f~ i 2 (f) ~ f(i 2 ) ~ 4{/(y) + (i 2 ) (j 2 ) + 2(y) 2 + 4i(ij 2 ) 

 + AjiPJ) - 2(ijij) - 4(i 2 j 2 ) (146) 



which by letting Aj and Bj be / of the second order should become 

 2A 2 s B 2 s ~ 2(A 2 )s(B 2 ) s as is easily verified. 



In simplifying, use is to be made of the identity 



{A i A 1 ) s = A iS A jS -A* i A j (147) 



where the last term is the star product of A i and Aj as already defined. 

 Since this scalar is an invariant of the two matrices it may well be 

 abbreviated A*^. Collecting and reducing results we find 



v U = A"\ s B"\s+ A" iS B" iS A* ii B* ii + A" iS A" jS B* 2 ,-,- 



+ ffWiaA** ~ 2A" lS A" lS B" iS B" jS + A" iS B" jS A*nB*i} 



+ A" 2 jS B" 2 jS (148) 



and by similar processes, 



m 3 = A i sA" i sB i sB" iS + A iS B jS B" i sA* ii + A iS B i sA ,, i8 B* ii 



— A jsB jsA" isB" is + AjsBjsA" jsB"js + AjsBisB"jsA*ns 

 + A iS B 3S A"isB* ij - A iS B iS A" jS B" jS (149) 



ma= A 2 iS B" iS - 2A" iS B" iS + A" iS B 2 iS + AigAjsB*^ - A*^*^ 



+ BiaBiaA+a + A 2 ]S B" lS - 2A" jS B" ]S + A" ]S B 2 ]S (150) 



hji= A iS Bis+ A ]S Bjs (151) 



whence (143) is completely solved. 



