- A71 Upper Limit for the Value of a Determinant. 
327 
limits are identical. The same also is true when the elements of fi are 
equimodular. 
8. When the elements of jjl are proportional to the corresponding elements 
of M', the elements of jj,' are proportional to the corresponding elements of 
M, and therefore the moduli of the elements of ^ are proportional to the 
moduli of the corresponding elements of M. It thus follows that when 
the elements of n are proportional to the corresponding elements of M' 
and the elements of (or M) are equimodular, then the elements of M (or 
ju) are equimodular also. By rationalising the denominator of each ratio 
in (X.), it is thus seen that when the elements of are proportional to the 
corresponding elements of M' and are equimodular, the product of any 
element of ji by the corresponding element of M is constant, or, in 
Sylvester's language, fi is " inversely orthogonal." Also, if jj, be 
"inversely orthogonal" and have equimodular elements, the elements 
must be proportional to the elements of M', and therefore by a preceding 
result I fx I must have its maximum value. The problem of finding 
inversely-orthogonal determinants is thus closely connected with the 
problem of finding determinants of maximum value. Any results, there- 
fore, obtained by Sylvester in his efforts towards a solution of the former 
problem deserve attention in the present connection, our scrutiny being 
all the closer because of the fact that his assertions are not always 
accompanied by proof. 
9. Taking the very special form of determinant which represents the 
difference-product of z, y, x, to, namely, the alternant \ z°7j^x'^iv^ ... (, let 
us inquire if there be values of z, y, x, zu, which make it inversely 
orthogonal. 
On multiplying each element of | z^y^xHo^^ \ by the corresponding 
element of the adjugate determinant we obtain the array 
yxio\y''x^w^\, -z \ y°x^w^\, z~ \y°xHu^\, - z^ \y''xMD^\ 
- zxw I z°xHv^ I, y \ z^x'^w^o |, -y^ \ z°x^w^ y^ \ z°xHo'^ | 
zyio I z°y^w^ \, -x\z°yHo^\, x^ \z°yHo^^\, -x^\z°yHu^\ 
- zyx I z°7j^x'^ i, 10 z'^y'^x^ \, -iv^ \ z°y^x5 |, w3 j z°y^x^ |, 
and the condition for inverse-orthogonalism is that all the elements of this 
array be equal. Now the equality of the first and second elements of any 
row of the array is tantamount to the vanishing of ^zyx, the equality of 
the second and third elements to the vanishing of ^zy, and the equality of 
the third and fourth to the vanishing of S0 : consequently z, ?/, x, lu must 
be the roots of an equation of the form w4 = a. Again, the equality of the 
elements of the first column is tantamount to the vanishing of 
{z^ - y^)/{z - (?/4 - x^)/{y = x), {x^ - tv^)/{x - iv) : 
so that, since z, y, x, to must from the nature of the problem be all 
