If the $n\times n$ matrix $M$ is decomposed into submatrices, $$M=\begin{pmatrix}A&B\\ C&D\end{pmatrix},$$ where $A$ has dimension $m\times m$, then the determinant of $M$ can be decomposed as $$\det M=\det A\det D+X.$$ The multinomial $X$ in the matrix elements of $M$ is contains $n!-m!(n-m)!$ terms.

In the $n=3$, $m=2$ example given in the OP, this gives for $X$ the four terms $$X=a_{13} a_{22} a_{31} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{11} a_{23} a_{32}.$$ Notice that the indices of $X$ follow Hancock's description.

So I would paraphrase the sentence highlighted in yellow as "Write down the determinant $D_n$ of $f$ and within that expression single out the product of the principal minors $D_m$ and $D_{n-m}$."

If the $n\times n$ matrix $M$ is decomposed into submatrices, $$M=\begin{pmatrix}A&B\\ C&D\end{pmatrix},$$ where $A$ has dimension $m\times m$, then the determinant of $M$ can be decomposed as $$\det M=\det A\det D+X.$$ The multinomial $X$ in the matrix elements of $M$ contains $n!-m!(n-m)!$ terms, for a general matrix $M$. If the matrix is symmetric, the number of distinct terms is less.

In the $n=3$, $m=2$ example given in the OP, this gives for $X$ the four terms $$X=a_{13} a_{22} a_{31} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{11} a_{23} a_{32}.$$ Notice that the indices of $X$ follow Hancock's description.

So I would paraphrase the sentence highlighted in yellow as "Write down the determinant $D_n$ of $f$ and within that expression single out the product of the principal minors $D_m$ and $D_{n-m}$."

If the $n\times n$ matrix $M$ is decomposed into submatrices, $$M=\begin{pmatrix}A&B\\ C&D\end{pmatrix},$$ where $A$ has dimension $m\times m$, then the determinant of $M$ can be decomposed as $$\det M=\det A\det D+X.$$ The multinomial $X$ in the matrix elements of $M$ is contains $n!-m!(n-m)!$ terms.

In the $n=3$, $m=2$ example given in the OP, this gives for $X$ the four terms $$X=a_{13} a_{22} a_{31} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{11} a_{23} a_{32}.$$

If the $n\times n$ matrix $M$ is decomposed into submatrices, $$M=\begin{pmatrix}A&B\\ C&D\end{pmatrix},$$ where $A$ has dimension $m\times m$, then the determinant of $M$ can be decomposed as $$\det M=\det A\det D+X.$$ The multinomial $X$ in the matrix elements of $M$ is contains $n!-m!(n-m)!$ terms.

In the $n=3$, $m=2$ example given in the OP, this gives for $X$ the four terms $$X=a_{13} a_{22} a_{31} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{11} a_{23} a_{32}.$$ Notice that the indices of $X$ follow Hancock's description.

So I would paraphrase the sentence highlighted in yellow as "Write down the determinant $D_n$ of $f$ and within that expression single out the product of the principal minors $D_m$ and $D_{n-m}$."