Dr Baker, On the Invariant Factors of a Determinant. 67 



sets of solutions of D 2 u = 0, say u {p+l) , ..., for which there exists no 

 relation 



H 1 Du^ + ... + H p DuW + H p+l Du^+v + . . . = 0, 



is q — p, a set of such q — p being it {p+1) , ..., u®. Then, supposing 

 the case when q = p to be included, each of the sets 



satisfies Dv = 0, and they are linearly independent. If beside 

 these there exist other solutions v {q+1) , ... of Dv = 0, such that there 

 exists no linear relation connecting v {1) , ..., v iq) , v {q+l) , ... let r—q 

 be the greatest number of these, and choose such, say v {q+l) , ..., v {r) . 

 Then v {1) , ..., v [r) are linearly independent; possibly q = r. 



Then we have the following f+ g + ... + r sets of solutions 



of D l x=0; x®, ...,xW, 



of D l ~'y = ; Dx {l \ ..., Dx^, y^+v, ..., yto, 



of D l ~ 2 z = 0; D°-xM, ..., DV, DyW +1 \ ..., DyW t z (ff+v ; ..., 



of D 2 u = 0; D l ~ 2 x^, ..., D l ~ 2 x^\ D l ^yV+», ..., D l ~ 3 yW, 



Dl-i z (g+i) ) mmmf J) t ( P ) ) u (p+i) 3 ...,«<«, 



of Dv = ; D^x.u, ..., D l - X x^\ D l ~ 2 y^+ l \ ..., D l ~ 2 yW, 



D l - 3 z(v+»,..., DHW, D«<p +1 >, ...,Du®, v (q+1 \ ..., v {r K 



§ 2. In regard to these sets of solutions it is easy to prove 

 (a) that they are linearly independent, (b) that their number is 

 equal to the number of linearly independent sets of solutions 

 of the equations D l x = 0. 



As to (a) we notice that if we suppose any linear relation, say 

 (A), to connect the sets, the result of operating on this relation 

 with D l_1 is to reduce it to a linear relation connecting 



D l - l xU,..., D l ~ x xV\ 



which, by the construction of the sets, can only be true when the 

 coefficients of x {1) , ..., x if) in (A) are each zero. The supposed 

 relation thus reduces to a relation, (B), connecting the sets other 

 than x w , ..., x^K By operating on this with D l ~ 2 this is again 

 reduced, and so on. 



As to (b), let x be any set of solutions of D l x = 0. Since / is 

 the greatest number of sets satisfying D l x = 0, for which a relation 

 of the form 



A,D l - x x^ + ... + AfD^x^ = 



5—2 



