388 HITCHCOCK. 



This fact will be fully exemplified below. A symbolic equation 

 implies, not the equality of individual terms each to each, but equality 

 of the sums of all possible terms of the equated forms. 

 Following a similar procedure to find ?» 3 we have 



6m 3 = ((<P, <p, <p))s= ((**>' I a ' b > cd ' I c ' d - ef ' I e ' f ))s, symbolically, 

 = aa'bb'cc'dd'ee'ff- aa'bb'[ce'c'e] [d'-fd-f] 

 - cc'dd'[ea'e'a] [f'bfb'] - ee'f-f'[a-c'a'-c] [b'dbd'] 

 + (ab' : c'd) (cd' : e'f) (ef' : a'b) + (ab' : e'f) (cd' : a'b) (ef' : c'd) 



(121) 



by direct application of the definition of Art. 5, and by grouping the 

 dot products in the second, third, and fourth terms after the manner 

 of (119). The first or leading term is evidently the same symbolically 

 as A i$B i$A jsB j$A ksB ks- The next three terms taken together are 

 symbolically — ZAisBi S (AjA k )s{B 3 Bk)s- In transforming the fifth 

 term the procedure, as in all cases, will be to group into one factor 

 those vectors which correspond to prefactors A in the original equa- 

 tion, and into another factor those which correspond to the post- 

 factors B. Thus 



(ab'rc'd) (cd':e'f) (ef':a'b) = (ac'ce'ea') (b'dd'ff'b) 



= (aa'ee'cc') s (bb'dd'ff') s 

 = (AiAtAMBiPiB k ) 8 (122) 



It is especially worthy of remark that the order of the matrices in one 

 factor is the reverse of that in the other, a consequence of the trans- 

 formation (112). When there are only two matrices in each factor, 

 as in (119), it was of no importance whether we wrote (AC)s or (CA)s, 

 for these are equal: the scalar of the product of two matrices is inde- 

 pendent of their order, because (aa'-cc')s = ac'a'-c = (cc'-aa'). 

 But by similar reasoning we see that the scalar of the product of several 

 matrices depends on their cyclic order, as is well known. 



In the same way the last term of (121) becomes (AiAjA k )s(BiBkBj)s. 

 Collecting results, the complete statement for m 3 is 



6ra 3 = A iS Aj S AksBisB jS Bks— A iS B iS (AjAk)s(BiBk)s — 

 AjsBjsiAkAMBkBih 

 - A ks B kS (A i AMB i B i )s+ (A i AkA ] )s(B i B j B k ) s + 



(AiAjAMBiBrfds (123) 



which is to be summed over all possible sets of three terms from the h 

 terms of the original equation (111), allowing repetitions for the reason 





