ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 365 



of p- row minors along the main diagonal of the determinant of the 

 in ik- By the usual matrix theory, it is therefore p!m p . 



In the above demonstration it is assumed that <p has been expanded 

 in the form (23). It follows from the distributive character of all the 

 steps involved in calculating these scalars that their value is inde- 

 pendent of the particular form of <p, precisely as in the case of similar 

 scalars occurring in the usual vector analysis. This is a property of 

 fundamental importance. For, when we have to perform operations 

 into which these quantities enter, it is frequently sufficient to treat a 

 single expression such as (35) in place of the sum of expressions ob- 

 tained by expansion of (36) ; and we may develop <p, when necessary, 

 as 2AM in any desired manner. 



To calculate any coefficient m p in the Hamilton-Cayley equation, 

 we have then merely to express <p as 2AM in any convenient way, 

 make all possible selections of p different terms, form the scalar (35) 

 for each selection, and add the results, dividing by pi. 



Many other analogies with ordinary dyadics will naturally suggest 

 themselves. It will be sufficient at this point to note briefly two of 

 the more significant of these. 



By solving the Hamilton-Cayley equation as if <p were a scalar we 

 may find roots g\, g- 2 , • • •, g n and corresponding polyadics Ri, R>, • ••, 

 R„ such that 



{fP -of) : Ri = 0, (i= 1,2,- --,n). (39) 



The polyadics Rr • R„ may by analogy be called the axes of the double 

 polyadic p with respect to dot product. 



Again, we may say that a double polyadic which can be expressed 

 as the sum of I (and no fewer) properly chosen dyads AM has n—l 

 degrees of nullity. To get the simplest criterion of the number of 

 degrees of nullity, we may, following Gibbs and Wilson, 6 introduce 

 cross products of polyadics by the law 



AXB = - BAA (40) 



and it is then possible to follow reasoning parallel to that of the paper 

 referred to in Note 3, so far as concerns the number of degrees of 

 nullity, by use of double powers of <p. Evidently a product defined 

 by (40) is not itself a multiple cross product in the sense that it would 

 reduce to double cross product when K — 2. For in Gibbs' system 

 the product by double cross of two dyadics with one common factor 

 vanishes, thus ab*ac = 0, while by (40) the cross product of two 

 polyadics vanishes only when they are scalar multiples of one another. 



