188 Transactions of the Royal Society of South Africa. 
and develop it according to products of a, h, c, d. The cofactor of ahcd is 
seen to be 1 ; the cofactors of ahc, abd, acd, bed to be 
f?i + ^?2 + 6?3, + + b,-]-b.-{-b^, ^2 + ^3 + ^^; 
the cofactors of ab, ac, ... to be two-Hne determinants which by hypo- 
thesis are unisignant ; the cofactors of a, b, c, d to be three-Une 
determinants which are also unisignant for the same reason ; and the 
aggregate of remaining terms, namely, those independent of a, b, c, d to 
be a determinant which vanishes because the sum of the elements of its 
every row is zero. The final development therefore consists of nothing 
but positive terms, the number of them being 
1 + 4-3 + 6-8 + 4-16, i.e., 125, 
or (4 + 1)-*"' in agreement with Sylvester's statement. 
In a sense, the result is a special case of the general theorem that if 
all the coaxial minors of a determinant be in their final development free of 
negative terms, any positive quantities added to the diagonal elements ivill 
not affect the unisignancy. 
4. Of much more importance, however, is the fact, strangely over- 
looked by Sylvester, and indeed not hitherto noticed, that the determinant 
obtained from his by changing the signs of all the non-diagonal elements is 
also free of negative terms in its final development. (II.) 
For this the same mode of proof suffices, but the final portion of it 
brings up a difficulty which is wanting in the other case, and which on 
investigation leads to specially interesting results. 
Denoting Sylvester's determinant when of the nth order by S„, and 
that now referred to by T„, we readily see that 
T2 = S2 = a6 -H aj) + ab^, 
T3 = S3 + ^{a^b^c^ + a^b^c^) ; 
that in T^ the cofactors of abed, abc, abd, acd, bed, ab, ac, ad, be, bd, cd 
are identical with the corresponding cofactors in S^ ; that the cofactors 
oi a, b, c, d are greater by 
2(6304(^2 -f 6462^^3), 2{ax^dj_-\-a^c^d^), 2{aJ)^d^-\-aJ)^d^), ^{a^b^c^ + a^b^c^) ; 
but that the aggregate of the terms free of a, b, c, d, namely, 
a2-{-a^-\-a^ a^ a^ a'^ 
bi b^ + b^ + b, ^3 b^ 
Cr C2 C^ + C,-\-C2 C^ 
d^ d^ d^ + d^ + d^ , 
