Two quick questions on notation, motivated by my being reading Cayley at the moment (I stumbled across a random volume of his Collected Works and now I am unable to do anything else but read it through—good libraries are the worst place for work!).

**I.** He uses the notation
$$(a,b,c,\dots)\!\!\!(X,Y,Z)^3$$
for forms (the parenthesis uses by the printer are more pronouncely curved than the ones mathjax is using here, so the crossing pair looks much nicer in the printed book than my poor rendition) The above is clearly a cubic form on $X$, $Y$ and $Z$ with coefficients $a$, $b$, $c$, &c. Can someone tell me

what ordering of the coefficients is used on the left of the $)\!\!\!($ ?

(This notation allows him to write the general polynomial of degree $n$ as $(a,\dots)\!\!\!(x,1)^n$, which is certainly nice!)

In some places, the parenthesis in $)\!\!\!($ which has its concavity to the left is adorned with an arrowhead on the upper end. I'd love to know what that means!

**II.** He writes determinants as in $$\begin{vmatrix}a,&b,&c,\\d,&e,&f\\g,&h,&k\end{vmatrix}$$ (with commas) but sometimes he writes things like
$$
\begin{array}{c}
\begin{pmatrix}a,&b,&c\end{pmatrix} \\
\begin{vmatrix}d,&e,&f\\g,&h,&k\end{vmatrix}
\end{array}
$$

What does that denote?

There are also a few instances of $$ \begin{array}{l} \begin{pmatrix}a,&b,&c\end{pmatrix}\!\!\!(x,y,z)\!\!\!(x',y',z') \\ \begin{vmatrix}d,&e,&f\\g,&h,&k\end{vmatrix} \end{array} $$ which presumably is a notation for a bilinear ternary form, combing the two notations...