## Certain notations in Cayley’s work

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...

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This is one of my first times where a telephone with a camera might have come useful! – Mariano Suárez-Alvarez Oct 26 2011 at 10:46

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.

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 $$ax^3+by^3+cz^3+3(fy^2z+gz^2x+hx^2y+iyz^2+jzx^2+kxy^2)+6lxyz$$

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 Yup. I did look there. I could not find the notation I there (if the first group $(a,\dots,...)$ were a (cubic) matrix, then I can see how to generalize the notation in page 94, but it seems to be a vector, so he flattened the coefficients somehow) The notation in II, without the $)\!\!\!(x,y,z)\!\!\!(x',y',z')$ attached I could not find. I do have to look in Muir's yet. – Mariano Suárez-Alvarez Oct 26 2011 at 12:11 See also page 99, and the footnote there, for II it is just a notation for a matrix multiplied by a vector. Cayley used two variants of this notation, one where the vector is on top of the matrix and one where it is next to the first row. As for I, I'm pretty sure the ordering is lexicographic, but I'll have to confirm that elsewhere. – Gjergji Zaimi Oct 26 2011 at 12:31 So, looking at Cayley's "An introductory memoir upon quantics", he says that in the (a,b,c,…)(X,Y,Z)^3 notation a term is not only multiplied by its corresponding literal constant but also by the corresponding multinomial coefficient. He uses a slightly different symbol for "unweighted" forms. On the other hand he also says that this notation is used "in generality" and never bothers to actually predescribe an order. – Gjergji Zaimi Oct 26 2011 at 13:07

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.

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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.

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