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



and satisfy no linear relation 



A.D 1 - 1 ^ + A^D^x® + . . . = 0, 



wherein A 1} A 2 , ... are each single quantities. Choose such sets, 

 say x {1) , ..., x {f) . Then each of the sets 



yW=DccU,..., yW = DxV\ 



satisfies the equations 



n i ~ l y = 0, 



and there exists no linear relation connecting the sets 



Uh*y®, ...,D l -*yV). 

 If beside y (l) , ..., y { ? ] there exist other sets of solutions 



y(f+D y ytf-M),..., of B l ~ 1 y = 0, 

 such that there exists no linear relation 

 B 1 D l -"yU+... + B f D i - 2 yV) + B f+1 B l -H/f+^ + B f+<2 B l - 2 y^^ + ...=0, 



in which each of B 1 ... Bf +2 ... is a single quantity, let g — /be the 

 greatest possible number of them, and choose such, say y(f + v ...y (g) . 

 Then, supposing the case when g—f to be included, each of the 

 sets 



z (D = Dy®,...,zW = DyW 



satisfies the equations 



D l ~ 2 z = 0, 



and there exists no linear relation connecting the sets 



D l ~ z z {1 \ ..., B l ~ 3 z {g) . 



If, beside z {1) , ..., z (g) , there exist other sets of solutions 



z (g+v } z^+*>,.,., of D l ~ 2 z=0, 



such that there exists no linear relation 



dIP+z® + ... + CglP-'zM + C g+1 D l - 3 z^+v + ... = 0, 



we can proceed as before. We shall arrive at length at sets of 

 solutions 



D l ~ 2 x^, ...,D l ~ 2 x^, D l ~ 3 x^+ l \ ..., D l ~ 3 xW, Ip-*ziff+v, ..., 



of the equations D 2 u = 0, which we denote in turn by 



tt«, ..., u { f\ u^ +1 \ ..., u [ 9\ u^ +1 \ ..., m<p), w'^ 1 ), ..., u ( ®, 

 which are such that, beside u {1) , ..., u^, the greatest number of 



