I recently read a paper by Cayley entitled "On the theory of linear transformations" (it can be found in volume I of his collected works as number 13), wherein Cayley computes a "hyperdeterminant" for a $p=3$, $m=2$, $n=4$, which is his was of saying he's computing a function in the entries of a $2\times 2 \times 2\times 2$ tensor that can be expressed as a homogeneous polynomial of degree 3 in the maximal minors of certain matrices listing all the entries of the tensor in question. More precisely, if we let $$a = 1111, b=2111, c =1211,d = 2211, e = 1121, f = 2121, g = 2211, h = 2221$$
$$ i = 1112, j = 2112, k=1212, l= 2212, m = 1122, n = 2122, o = 2212, p = 2222$$
the matrices in question (whose maximal minors we're interested in) would look something like (they're only well defined up to a permutation of their columns):
$$M^{(1)} = \begin{pmatrix}
a & c & e & g & i & k & m & o \\
b & d & f & h & j & l & n & p \\
\end{pmatrix}\:,\:
M^{(2)} = \begin{pmatrix}
a & b & e & f & i & j & m & n \\
c & d & g & h & k & l & o & p \\
\end{pmatrix}$$
$$M^{(3)} = \begin{pmatrix}
a & b & c & d & i & j & k & l \\
e & f & g & h & m & n & o & p \\
\end{pmatrix}\:,\: M^{(4)} = \begin{pmatrix}
a & b & c & d & e & f & g & h \\
i & j & k & l & m & n & o & p \\
\end{pmatrix}$$
where the first matrix lists all possible columns of the form $\begin{pmatrix}1rst\\2rst\end{pmatrix}$, the second lists all columns of the form $\begin{pmatrix}r1st\\r2st\end{pmatrix}$, and similarly for the third and fourth matrices with the third and fourth indices, respectively, where we always have $$1\leq r,s,t\leq 2$$
To recap, Cayley's definition of a hyperdeterminant of order $p$ of a tensor is a polynomial in the coefficients which can be written as a (usually distinct) polynomial homogeneous of degree $p$ in the maximal minors of each of the matrices listed above (these matrices obviously being only for this particular case). For the case in question, Cayley writes:
and the stated values continue for another half page or so. It's not clear to me what exactly Cayley is computing, and how. If someone could enlighten me, I would greatly appreciate it.