192 Proceedings of Royal Society of Edinburgh. [sess. 
material for proving the two theorems above referred to. The first 
of them, which is not new, is, in later phraseology that “ Any zero- 
axial skeiu determinant of odd order vanishes ” ; and the second, 
which is Cayley’s own, is that “ Any zero-axial skew determinant of 
even order is the square of a Pfaffian.” In both cases the method of 
proof is that known as ‘ mathematical induction ’ ; and in both 
cases the main auxiliary theorem used is Cauchy’s regarding the 
expansion of a determinant according to binary products of the 
elements of a row and the elements of a column. 
When n is odd and the elements of the first row and those of 
the first column are 0,A. 12 ,A. 13 , . . . , \ in and 0,X 21 ,X 31 , . . A W1 
respectively, he says it is easy to see that for each term having 
AiaA/ 3 i f 1 or a factor, where a # /?, there exists an equal term of 
opposite sign having A.ijsA a i for a factor ; and that therefore, since 
AiaA/si = Ai^Aai, these two terms must cancel each other. As for 
the terms which have Ai a A«i for a factor, the co-factor is a deter- 
minant of exactly the same form as the original, but of the order 
n- 2 ; consequently the theorem is seen to hold for any one case 
if it hold for the case immediately preceding. But for the case 
where n — 3, the theorem is self-evident ; therefore, “ Tout deter- 
minant gauche et symetrique d’un ordre impair est zero.” 
When n is even, the determinant dealt with is purposely taken 
more general than one with skew symmetry, although, strange to 
say, Cayley calls it ‘ gauche et symetrique,’ the elements of the 
first row and those of the first column being Kp,K 2 ,Ks, . . . , Kn 
and A a/3 ,A 2 / 3 ,A 3 y 3 , . . . , A Wj3 , and his aim being to prove that such a 
determinant is equal to the product of two of the functions treated 
of in the digression, viz., [a 2 3 ... n\ and [/3 2 3 . . . n\. Develop- 
ing as in the preceding case, there has this time to be considered 
the element common to the first row and first column, viz., 
A« 3 , the co-factor of which is seen to be a skew symmetric deter- 
minant of odd order n — 1, and therefore, as has just been shown, is 
equal to zero. As for the co-factor of - Kn'hpp, where \ aa ' is any 
element of the first row except the first, and is any element 
of the first column except the first, it will be found to be a deter- 
minant which Cayley again mistakenly but consistently calls 
c gauche et symetrique,’ obtained by giving to r all the values 
2,3, ... i ft with the exception of a', and to s all the values 
