174 WILSON: REDUCTION OF REAL DYADICS 



have no connection with Grassmann's Erganzung.) The only 

 kind of multiplication used in the work is the (progressive or 

 regressive, single or multiple) combinatory product. That 

 means that we work with the affine, not with the metric, group 

 of linear transformations with one point fixed, i. e., with homo- 

 geneous strains. 



2. The essential steps in the reduction of a dyadic are these: 



(a) As there are only n 2 independent dyads, any dyadic must 



satisfy a polynomial equation, with numerical coefficients, of 



degree not exceeding n-. Hence any dyadic satisfies an equation 



A O) = $ m + ci$ w_1 + . . . + c m _ x 3> + c m 7=0 



of lowest degree, I being the idemfactor. This equation is unique 

 for if there were two of like lowest degree, their difference would 

 be of lower degree (pp. 15-16.) 



(6) As any dyadic is homologous (commutative in multipli- 

 cation, p. 14) with its powers and with I, the equation of lowest 

 degree may be factored in the form: 



A ($) = ($ - aiy ($ - biy ($ - ciy . . . = o 



identical with that obtained in factoring the ordinary polynomial 



A (x) = x m + d x m ~ l + . . . + C m _! X + c m 



= (x - a) p (x - b) q (x - c) r . . . =0. 



(c) If we set (p. 26) 



$ - al = ¥, * - bl = ¥ + (a - b) I, <J> - cl = * + (a - c) I, . . . 



and if we divide (by the ordinary algorism of long division, 

 which is applicable here) the expression 



(?-bI) q (*-cI) r . . . =AI+B*+ . . . +H* q+r+ --- 



into I, we have the result 



^I + B'^ + .-. + G'^- 1 





+ 



AI +B* + ... + HV n - p 

 where P is a polynomial of degree m — p - 1. Next, let (p. 27) 



