Dr Baker, On the Invariant Factors of a Determinant. 77 



must not be less than 



I - (k + fi + . . . + o) + 1 + V + I" + . . . 



and not greater than 



Putting n = I + l' + l" + ... we further see immediately that the 

 powers of © arising in the various powers of R give rise to a 

 table obtainable from that above, for the powers of Q, by the sub- 

 stitution only of R for Q and n for I. 



Comparing this result with the expansion given, § 7, for 

 \<f> + tb\ we deduce at once, beside the result 



I = ay + • • • + /-t/3 -f tea, 



that 



l^a = € 1 = e 2 =... = e K >8 = e K+1 = . . . = e K+ll > . . . > 7 



while e K+M+ ... +<r+1 . . . are all zero, so that the rank of the matrix 

 a — p in regard to p — 6 is n — {tc + . . . + a). 



§ 9. The result then is that for any root 6 we have rows of 

 equations of the form 



(a — 6) x x = 0, (a — 6) x 2 = x x , . . . , (a — 6) x e = x e _ x , 



there being one such row corresponding to each of the invariant 

 factors associated with that root, and the n sets denoted by x 

 which arise can be chosen to be linearly independent. 



