356 HITCHCOCK. 



A dyadic is a sum of dyads, or, with no increase in generality, a sum 

 of dyads each multiplied by a scalar factor. The elements of a dyadic 

 are the sums Zctibk of corresponding products taken from each dyad 

 term, where, for an element Aik of a dyadic A the subscripts i and k 

 are constant and the summation is over the various dyad terms. 

 Thus if 



A = aibi + a>b 2 + • • • + a A b A (2) 



we have 



Aik = aubik + 02ib 2 k + • • • + (ihibhk . (3) 



It is frequently of use to write the dyad ab as symbolical of the dyadic 

 and dibk as symbolical of the element Aik, omitting the subscript h 

 and the sign of summation with respect to h. 1 



A dyadic may be written in terms of the dyad units efik thus 



A = 2A ik 9ifi k (4) 



where both subscripts run from 1 to A r . 



In a similar manner several vectors abc • • • g written with no sign 

 between them constitute a polyad. When we wish to indicate that 

 the polyad is of order A, that is, it is the indeterminate product of 

 K vectors, we may call it a A-ad. The vectors a, b, c, etc., in order, 

 will be called the first, second, third, etc. factors of the polyad. A 

 polyadic is a sum of polyads, all of the same order. The polyads 

 e p e q e r - ■ e s to K factors will be called the fundamental polyads of 

 order K or fundamental A-ads. It is evident that any polyadic may 

 be expanded in terms of the fundamental polyads of its own order. 

 The elements of a polyadic are the scalar coefficients in this expansion. 

 Thus if we have K subscripts p, q,- • • , s, all of which run from 1 to X, 

 we may write a A-adic A as 



It is thus apparent that in general a Zv-adic depends on X K scalar 

 elements. These may be primarily regarded as forming a A-dimen- 

 sional block, and most conveniently written by means of adjacent 

 squares when special values have to be assigned. For example if 

 X = 2 and A = 4 we should have the scheme of elements 



^4im, -bir: 

 Ami, Aims 



Amu Ann 



Al221) Ai22S 



A2UI, A 2112 

 -42121, -42122 



• I22IIJ -4.2212 

 A232I) A2222 



