1899 - 1900 .] Dr Muir on the Theory of Skew Determinants. 207 
determinant P is a square (“nella quale equazione trovasi appunto 
espressa la propriety che il determinante P e un quadrato ”). 
On looking now to the development with which- the demon- 
stration opened Brioschi is led to an expression for the square 
in question, viz. : 
or, more generally, 
where he notes that in every case a rr — 0 and 0 2 P /da rr da ssf being a 
determinant of the same kind as P, is a square. The example 
added is 
0 
a l2 
<*13 
<*14 
a 21 
0 
<*23 
<*24 
f 
0 a M 
i 
0 a 24 
i 
0 a 23 
<*31 
<*32 
0 
<*34 
II 
JP 
to 
a 43 0 
- a 13 
<*42 ^ 
+ <*14 
<*32 0 
} 
i 
<*41 
<*42 
<*43 
0 
— (a i2 $ 34 ^13^24 + <*14 a 2s) 2 } 
where the difficulty of the ambiguous sign, although presenting 
itself more prominently than in the general demonstration, is not 
referred to. 
The new function H, which is the square root of P, is next 
studied. Differentiating both sides of the equation of relationship 
Brioschi obtains 
0P = h 0H * 
0«M 9 <*r S ’ 
where the inconvenience of the differential notation comes out 
more strikingly than before, the differential-quotient on the left 
being used conventionally to denote a certain minor of P, and 
the differentiation on the right being real. By squaring we have 
I QJ 
1 
to 
II 
hd 
/0H \ 2 
\da rs / 
\0a„/ 
* Since the left member is what Cayley called a “bordered skew sym- 
metric determinant ” ; and since, as Jacobi noted, a differential -quotient 
of H with respect to one of its elements is a function of the same kind 
as H, we have here one half of Cayley’s proposition that a bordered skew 
symmetric determinant is expressible as the product of two Pfaffians. 
