Certain notations in Cayley's work

Question

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...

Answer 1

Some of the symbols used in that time would be tricky to type in Latex, so instead of writing an explanation here, I hope it is okay to just give a reference. There are two books by Florian Cajori, "A History of Mathematical Notations: Vol. I and II", which are a good reference for this specific kind of question. Cayley's and related notations on determinants and n-ary forms can be found in vol II, starting at page 87, particularly look at page 94.

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As mentioned in the comments, the book above discusses only the notation for bilinear forms. For n-ary forms Cayley gives a more explicit description of his notation here page 413: $$(a,b,c,f,g,h,i,j,k,l)(X,Y,Z)^3$$ with the right symbol instead, stands for $$ax^3+by^3+cz^3+3(fy^2z+gz^2x+hx^2y+iyz^2+jzx^2+kxy^2)+6lxyz$$

Answer 2

For binary forms a notation such as $$ (a_0,a_1,\ldots,a_n)\!\!\!\!(x,y)^n $$ means $$ a_0 x^n+ a_1 \left( \begin{array}{c} n \\\ 1\end{array} \right)x^{n-1}y+ a_2 \left( \begin{array}{c} n \\\ 2\end{array} \right)x^{n-2}y^2+\cdots+a_n y^n $$ If Cayley uses the notation with the pointy arrow on one of the parenthesis he means the same thing without the binomial coefficients. For $p$-ary forms, I believe there must be a choice of ordering of monomials hopefully specified in the paper under consideration.

Answer 3

In answer to II, I have a Penguin book called The Treasury of Mathematics which reprints Cayley's "A memoir on the theory of Matrices". In it he says

"...the term matrix used without qualification is to be understood as meaning a square matrix; in this restricted sense, a set of quantities arranged in the form of a square, e.g.

$$ \begin{array}{c} \begin{pmatrix}a,&b,&c\end{pmatrix} \\\\ \begin{vmatrix}d,&e,&f\\\\g,&h,&k\end{vmatrix} \end{array} $$

..."

So it looks as if that is to be interpreted simply as a 3x3 matrix. He then says that the 3x3 system

$$\begin{eqnarray*} X &= ax+by+cz \\\\ Y & = a'x + b'y + c' z \\\\ Z &= a''x + b'' y + c '' z \end{eqnarray*} $$

"may be more simply represented by"

$$ (X,Y,Z)= \begin{array}{l} \begin{pmatrix}a,&b,&c\end{pmatrix}\!\!\!(x,y,z) \\\\ \begin{vmatrix}a',&b',&c'\\\\a'',&b'',&c''\end{vmatrix} \end{array} $$

Somehow I've broken the formatting on that last one, but it is supposed to look like your final displayed equation without the $(x'\ldots $ bracket.