Axisymmetric Determinants. 447 



it follows that 



a = {(li) 4TT] + (12) Vp] + ... + {in) Vm}. Vtii], 



and therefore that 



(11) v/jTX] + (12) V[22] + ... + (In) s/[n~n~] = 0, 



it being impossible that the vanishing of A could necessitate the 

 vanishing of [11]. We thus reach the following general theorem : 

 If [rr] be the cofactor of the element (rr) in an axisymmetric deter- 

 minant A of the nth order, then the norm of 



( n/[TT], s/[22], ..., V[~?H}[ an y row of A ) 



is divisible by A. (I.) 



Sylvester's result is the particular case of this where the diagonal 

 elements of A are zeros, and the elements of the last row and of the 

 last column are units. 



3. In examining the character of the cofactor of A in the norm 

 let us confine ourselves for the present to the cases where there are 

 not more than three non-zero elements in the first row of A, and 

 first let us see what happens when all the elements of the first row 

 vanish except the last two. The norm then is 



N {(1, 7i-l) sj[n-l, ?i-l] -f (1, n) J[n, %]j 



and therefore is 



(1, n-l) 2 [n-l, n-1] — (1, n) 2 [n, n]. 



But in any determinant ivhich has (1, r) and (r, 1) equal to zero for 

 all values of r except n-1 and n, the cofactor of (w-1, n-1) is 



- (1, n) (n, 1) . | (2, 2) (3, 3) ... (n-2, n-2) \, 



and the cofactor of (n, n) is 



- (1, n-1) (n-1, 1) . | (2,-2) (3, 3) ... (n-2, n-2) \ ; 



so that the ratio of the two cofactor s is 



(l,n)(n,l) : (1,7^1)^-1,1). (II.) 



This ratio in the case of an axisymmetric determinant is 

 (l,ny : (1, n-1) 2 , which is the same as saying that the norm in 



