Dr Baker, On the Invariant Factors of a Determinant. 75 



Finally, as in the notation of § 5, form, with the same ©, a 

 symbol 



Let Q = Pa >K + iV+ •■•+-*%,*• 



Then, for the other factors W = p-6', ®" = p - 6", ... of the 

 determinant \b\, form corresponding symbols Q\ Q", .... 



We are to investigate the powers of © which enter as factors 

 in the various powers 



(Q + Q'+Q" + ...) s . 



§ 8. To do this we distinguish the powers of any such 

 symbol as useless, significant or vanishing powers, according as 

 they are divisible by no power of ©, by such a power, or vanish 

 entirely. 



Then firstly for 



W m = © (e x + ... + e m ) +f 2 + ... +f m 



we find that W m , W m 2 ,..., TF"™" 1-1 are useless powers, that W m m 

 is a significant power and divides by © m , and that 



W rn+i W m+2 



" m > ' ' m j • • • 



are all vanishing powers. For example, taking m = 3, 



W z = © (e, + e. 2 + e 3 ) +f 2 +f 3 , 

 gives W 3 2 = 2© 2 (e 2 e 3 + e 3 e 1 + e^) + 2© {e 1 f z + e 3 f 2 ) + 2/ 2 / 3j 



Wi^m z e x e 2 e 3 , 



F 3 4 = 0, 



Thus, taking 



a power of P a<K cannot be significant if its expansion contains 

 such a term as 



t j . ... r K . 



where everyone of the exponents r lt . . . , r K is less than a ; therefore 

 its lowest significant power is not less than k (a — 1) + 1. Con- 

 versely if we take this power one at least of the exponents 

 rj , . . . , r K in any term of the expansion is as great as a. Thus this 

 is the lowest significant power. The highest significant power is 

 clearly ica, and all higher powers are vanishing powers. We easily 



