382 HITCHCOCK. 



are normal and orthogonal under star product. As already noted, 

 the star product in this special case of two dyadics <p and 6 is the scalar 

 (<PxP)s of Gibbs. If we wish to restrict ourselves to self-conjugate 

 dyadics we have merely to drop F 7 , F 8 , and F 9 from the list. It would 

 be interesting to work out the analogy between the scalars called 

 above ((ip p ))s, which might be symbolically written ((ab)) p s, and the 

 invariants of ternary quadrics written in the usual symbolic language. 

 We see however that systems orthogonal under star product will 

 always be possible, because to every double polyad E,E S corresponds 

 always at least one other double polyad not orthogonal to the first. 



13. Scalars formed by Star Multiplication. 



In what follows I shall, for simplicity of language, take K = 1, so 

 that double polyadics become dyadics, and polyadics become vectors. 

 This imposes no formal limitation, and can be extended to the most 

 general case with slight verbal change. 



Supposing N to have any value, we assume Fi, F 2 ,- • • F y a set of 

 linearly independent dyadics orthogonal under star product. Any 

 other dyadics A, M, B, or N can be expressed in terms of these. 

 Let <pi and <^ 2 as before be a pair of double dyadics 2AM and 2BN. 

 The star product of AM into BN is defined to be A(M*B)N which is 

 the same as ANM*B. Thus <p*<f>2 is a new double dyadic. If we 

 regard v>V and <p*<p*<p etc. as a new type of power abbreviated cp* 2 , 

 <p* 3 , etc. no further proof is required to show that <p* satisfies a Hamil- 

 ton-Cayley equation similar in form to (28), 



<p*"- mi*^*C- 1 >+ •••+ (-l)m*J"= (89) 



nor to show that the star idemf actor I" is identical with 2F,F{. The 

 whole of Art. 3 might be repeated at this point, replacing multiple dot 

 product by star product and E by F. The coefficients m* p in (89) 

 will be formed as in Art. 5, replacing multiple dot by star product. 

 We shall also have a set of scalars 



(<Pi, <P2,' • • , <Pp)s* (9°) 



analogous with (36). For example 



(AM, BN)/= A*MB*N - A*NB*M (91) 



which differs from (32) in having star in place of colon. It also differs 

 in having no analogue when A, M, B, N are mere vectors, whereas 



