390 HITCHCOCK. 



scripts in the postfactors B is the same as in the polyadic term, but the 

 order of subscripts in the prefactors A is the reverse of the order in the 

 postfactors B. For in the polyadic expression b' s follows its mate b s , 

 and b' ( follows b ( , and if we rewrite the factor in the form (b r bV 

 bsb's-b/b'^s then b', also follows its mate b r , whence directly the 

 matrix factor (B r B s B t )s- But in the polyadic expression as it stands, 

 each a', when we follow the same steps, precedes its mate on account 

 of the reversal of the order of accents. Thence follows the reversal 

 of order of matrices in the corresponding factor. Transformations 

 of this type already occurred in the last two terms of ?»3- 



All the italicized statements in the above discussion are true no 

 matter how many consequents have changed places among themselves. 

 We may therefore write a general rule for the formation of m p . 



Rule for forming p!m P where m p is the coefficient of 6 P in the 

 Hamilton-Cayley equation for 6. 



Let there be p subscripts i, j, k, • • • , r, s, t, each of which may have 

 any value from 1 to h. Choosing a particular set of values for these 

 subscripts, we form a group of pi terms as follows: the leading term 

 is AisBisAjsBjsAksBk-s- ■ -^rsBrsAssBssAtsBts- The other terms 

 are formed from the leading term by first interchanging the post- 

 factors B in all ways, while the prefactors A are at first left fixed in 

 position. If a particular postfactor B is left in position, it yields in 

 the corresponding term a factor Bs and its pref actor yields As pre- 

 cisely as in the leading term. If a pair of postfactors as B r and B q 

 change places, there results in the corresponding term a factor (A r A g )s 

 (B Q B r )s- If three postfactors, as Bk, B r , and B„ change places so that 

 their new order is B r , B s , Bk, there results in the corresponding term a 

 factor (AkA s A r )s(B r B s Bk)s where the order of prefactors is the reverse 

 of the final order of the postfactors. In general if any group of post- 

 factors change places among themselves so that their final order is 



BkB r - • -B s Bj 



there results in the corresponding term a factor (A jA s • • • A r A k)s 

 (BkB r - • -B s Bj)s where the order of prefactors is the reverse of the 

 final order of postfactors. 



The group of p! terms thus obtained is to be summed over the extent 

 of the linear matrix equation. 



18. Character of the Coefficients as Algebraic Polynomials. 



It is evident from the form of the leading term that in every term 

 of the expansion of m p will be found p prefactors and p postfactors, 



