ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 361 



Each of the consequents M» may be expanded in terms of the Ej by 

 aid of the notation 



M f = wiaEi -f m i2 Eo 4 \- m in E n . (24) 



If the Mi of (23) be expanded by (24) the complete development of <p 

 will be in terms of the double polyad units EiE^- or 



<p = 2>» a -E t E,, [i, k, = 1,2,-" n]. (25) 



It is clear that a double polyadic depends on n", that is N 2K scalar 

 elements. Thus double dyadics in ordinary space, of which the idem- 

 factor (22) is a special case, would yield in general 81 terms if it were 

 necessary to expand completely, which fortunately it is not. A 

 double dyadic is of course a tetrad ic. If need arose to emphasize the 

 2/v-adic character of a double polyadic and at the same time to depict 

 the scalar elements, use could be made of a system of adjacent squares 

 like (6). In the applications which follow, however, the scalar ele- 

 ments of a double polyadic are to be thought of as a binary assemblage 

 or square array. For example if K = 2 and N = 2, with i and j for 

 unit vectors, the order of the fundamental dyads may be agreed upon 

 by taking E x = ii, E 2 = ij, E 3 = ji and E 4 = jj. The corresponding 

 scalars m pq will then be arranged as 



m n , W12, mi3, m u 



>»21, m 2 2, W23, ?»24 (26) 



m, 3 i, m 32 , rttzz, w? 34 



Vli\, 7» 42 , ?/?43, ?»44 



where m pq is the coefficient of E p E g in the double dyadic 2?n P gE p E g . 

 Comparing with the four-subscript arrangement (6), we see that the 

 upper left hand square of (6) would correspond to the first row of (26), 

 the upper right hand square of (6) to the second row of (26), and so on. 

 Returning to our fundamental concept of a double polyadic as a 

 sum of dyads whose antecedents and consequents are polyadics, we 

 next define the A'-tuple dot product <pii<p2 of two double polyadics, 

 (in precise analogy with Gibbs' dot product of two dyadics), to be the 

 sum of all terms of the form AM : BN or (M : B) AN obtained by mul- 

 tiplying out the two double polyadics and taking the /v-tuple dot 

 product of each consequent of ^i into each antecedent of <p 2 . Thus 

 (pi: <p-2 is also a double iv-adic. We abbreviate <p: <p as cp 2 , and cp: <p: <p 

 as <p 3 , etc. This multiplication is easily seen to be associative, as well 

 as distributive, but of course not generally commutative. 



