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



where H, H 1 , ..., H r ^ are integral polynomials in p, and ■yjr is an 

 aggregate of terms not divisible by © ; on the other hand 



l#+*l-l*l+«(sa, g y w+ '( ssl *8s) , ' l * l+ - ; 



hence we see that the determination of the numbers I, e lt e 2 , ... is 

 equivalent to the determination of the powers of © = p — 6 which 

 enter into the successive quantities 



Such powers arise in virtue of the fact that every term in the 

 expansion of this quantity which involves a factor 



d 2 



wherein either i = i x or j=j ly vanishes identically; and if we 

 neglect the possibility of the existence of linear relations con- 

 necting minors of the same order of the arbitrary matrix <£, they 

 arise for no other reason. Taking account of the special form of 

 |6j, our problem is then the same as the following: — 



Let, as before, © = p — 9 and 



W a = © (e, + e, + . . . + O + f % + f 3 + . . . + f a 



wherein e x . . . e a ,f 2 . . .f a are indeterminates obeying the fundamental 

 laws of algebra and such, in addition, that 



namely their squares, and the products of an e with an / in which 

 either the suffixes are the same or that of the f is greater by one 

 than that of the e, are all zero. Take k such symbols 



w (i) w (2) w M 



"a > ''a > • ■ • j "a j 



the e x ... e a , f 2 ■•■fa occurring in W a {i) not being the same as those 

 in WJti, and there being no equations such as e s {i) f s (i) = 0. Put 



Similarly form, with the same ©, a sum of fi symbols Wp, each 

 involving fi symbols e and fi — 1 symbols f say 



the symbols e, /which enter in any Wp {ii are subject to rules such 

 as those above, but the k + p, symbols W contain different sets of 

 such e and f And so on. 



