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MATHEMATICS: F. L. HITCHCOCK Proc. N. A. vS. 
= S((a,b^a^b,va;,b>',b,))5 
= S[(a,b^:a',b,) (a,b',: a',b,) - (a,b',: a', b,) (a,b',:a^b,)] 
= S[a,.a^bi:b>jfe.a';feb;fe.b'fe - eii.Si'kSik'ei'ihk.h'ihi.h\] 
= i:[AisBisAksBks - {AMs(BkB,)s] (10) 
where, as before, Af^ is the sum of elements of the main diagonal of Aj,. 
etc. and (AiAk) 5 is the sum of elements of the main diagonal of the product 
of the two matrices Ai and Ak. 
By similar transformation we may show that 
6W3 = i:[AisBisA,sBjsAksBks - AsBis{AjA,)s{BkBj)s 
- AjsBjsiAj,A,)s{BiBk)s - AksBks(A,Aj)s{BjBi)s 
+ {A,AjA,)s{BkBjB,)s + (A,AjA,)s{BiBjBk)s]. (H) 
In every case all subscripts run from 1 to h, and terms where some or all 
of the subscripts are equal do not in general vanish. For a matrix term 
AxB is not in general equivalent to a single dyadic term MN:X, 
The general rule for any m is now easy to see. The quantity pfnip 
may be written as a summation of square bracket expressions, each square- 
bracket enclosing p! terms. The leading term is 
^AisBisAj<^BjsAksBks • • • A^sB^s 
where the p subscripts i, j, k, . . r are those of the given matrix coefficients 
in (2) and may be either alike or dijff^erent. The other terms are formed by - 
first keeping the antecedents A fixed and making interchanges among the 
consequents B. A multiple interchange such that q consequents BoB^^Bc . . 
BgBfBg {whether alike or not) change places among themselves leads to a 
product of two factors as (A^AjjAg . . . AdAfAa)s {BaBfB^ . . . BeB},Bc)s where 
the order of consequents is the order they stand after the interchange, while the 
order of antecedents is the reverse of that of the consequents. Any antecedents 
or consequents which have not been moved yield the same scala^s as in the leading 
term. 
These quantities m may be transformed in many ways. 
6. A Second Method of Solution. — Another method of solving the equa- 
tion (2) follows at once from the analogy between d and an ordinary dyadic. 
If <^ = SNM let if' = SNM corresponding to 6', that is to i:A'xB'. 
Let Ci, C2, • ' • be n-i matrices or dyadics such that each is orthogonal 
to the given matrix C in the sense that C:Ci = 0 fori = 1, 2,. . . , n — 1. 
Form n — 1 new matrices B'{Ci) . A matrix orthogonal to all these new matrices 
will in general be a solution of (2) aside from a scalar factor which may be 
found by direct substitution. This scalar may also be found as a function 
of m„ and the matrices C, Ci, C2, . . . 
This method is analogous with Hamilton's original method of solving 
the linear vector equation.^ 
7. Conclusion. — The analogy between a dyadic and a vector by virtue 
of the double dot product may be extended to polyadics and i^-tuple dot 
