212 Proceedings of Royal Society of Edinburgh. [sess. 
as Brioschi did, to the case where A is zero-axial skew and of odd 
order to obtain the result 
A 2 ,, = A r) :A ss ; 
and he takes the further step of deducing from it the result 
Ayj ‘ A^2 A r , : . = J A-q : A.^2 • >/ A -33 • • • • 
thus showing, as he says (1) that the ratios on the left are inde- 
pendent of r, and (2) that, when the sign of one of the roots has 
been fixed, the others are known (“dass durch das Zeichen einer 
unter diesen Wurzeln die Zeichen der iibrigen Wurzeln bestimmt 
sind.”) 
Turning now to the section specially set apart for the considera- 
tion of skew determinants, we find that it opens with Cayley’s 
theorem regarding a zero -axial determinant of even order, the 
requirement being, as here worded, to prove that such a determinant 
is the square of a rational integral function of the elements. 
The proof is essentially the same as Spottiswoode’s and Brioschi’s, 
and differs from Cayley’s merely in that it does not begin with a 
determinant of a more general form than is necessary, — a point 
which it is desirable to insist upon, as Baltzer ignores the fact, and 
then does not hesitate to say in a footnote that Cayley’s proof 
“ leaves manifold doubts unrelieved.” In fact the theorem which 
Cayley proves is, that if a zero-axial shew determinant of odd order 
be ‘ bordered ’ the resulting determinant is the product of two 
Pfaffians : whereas what the three others prove, is the particular 
case of this in which the skewness extends to the bordering 
elements. 
The development with which the proof begins Baltzer writes in 
the form 
A — ^nAjj — A rs , 
rs 
where A' is the cofactor of a rs in A n , and r and s have the values 
2, 3, . . . , n. He then uses the fact that A n is a zero-axial 
skew determinant of odd order, and that therefore by a preceding 
result 
A rs = A sr — f A. rr A. ss y 
