# Unclear passages in the paper “On a New Class of Theorems in Elimination Between Quadratic Functions” by J. J. Sylvester

I'm writing an essay about the origin of some mathematical terms in the work of J. J. Sylvester. He first used the word matrix in his paper Aditions to the Articles "On a New Class of Theorems" and "On Pascal's Theorem".
I have a question about the original paper On a New Class of Theorems in Elimination Between Quadratic Functions (1850). It can be found in his collected papers (p. 139):
http://archive.org/stream/collectedmathem01sylvrich#page/n155/mode/2up
It's 6 pages long.

This is what I've got so far:

1. The main theorem he gives is:
Let $U$ and $V$ be respectively quadratic functions of the same $2n$ letters. Then the following statements are equivalent:
• It is possible to institute $n$ such linear equations between these letters as shall make $U$ and $V$ both simultaneously become indenticaly zero.
• The determinant of $\lambda U+\mu V$ is the square of a function of $\lambda$ and $\mu$.
2. He uses this notation for a determinant: $\underset{x, y}\square$
3. His determinants have the opposite sign than the ones we use.

Now to my question:
As an example, he uses the functions \begin{eqnarray*} U& = &xy+yz+zx+(lx+my+nz)t\\ V& = &cxy+ayz+bzx+k(lx+my+nz)t, \end{eqnarray*} where $x, y, z, t$ are variables and $a, b, c, k, l, m, n$ are parameters. He writes:

A geometrical demonstration may be given of the theorem... The equation $$\underset{x, y, z, t}\square [\lambda U+\mu V+(lx+my+nz)t] = 0,$$ which is a quadratic equation in $\lambda : \mu$, may easily be shown to imply that the conic $\lambda U+\mu V$ is touched by the straight line $$lx+my+nz = 0.$$

How can the equation above define a straight line and how can the expression $\lambda U+\mu V$ define a conic? Does he use some notation different from ours or is it a mistake? Is there maybe some relation between the variables that I missed?

UPDATE: I think the fact that there are too many variables in the equations of the conics and the line has something to do with what Sylvester calls "indeterminate coordinates". He writes in a different article:

The indeterminate analysis assumes at will any number of coordinates, and leaves the relations which connect them more or less indefinite, and reasons chiefly through the medium of the general properties of algebraic forms, and their correspondencies with the objects of geometrical speculation.

But how exactly does it work?

• Capitalization of titles are not necessary, (which has a meaning that you needed to shout) – Narasimham Jul 23 '16 at 18:03
• YES. WHAT @NARASIMHAM SAID. I MEAN, DO YOU FEEL THIS COMMENT IS PAINFUL ON THE EYES? – Asaf Karagila Jul 23 '16 at 18:06
• Point taken. I just wanted to differenciate the titles from the body of the text. And also, the original titles of the articles are written in caps... – Mikulas Jul 23 '16 at 18:56