## 4-18-2. Arthur Cayley to H. Poincaré

Cambridge 3^{rd} March 1884

Dear Sir,

I have to thank you very much as well for the last paper in the
*Acta* as for the very interesting Notice of your Scientific
works.^{1}^{1}endnote:
^{1}
Poincaré (1882, 1884). I am at last at leisure to
do so, and I am beginning to study your beautiful theory. But I am
puzzled with your example I, of the
quadrilateral (14, 23).^{2}^{2}endnote:
^{2}
Poincaré (1882, § 7). Taking a
symmetrical form $ABCD$ as shown in the figure, and applying to it the
transformation which converts $AB$ into $DA$, I obtain a new form
$A_{1}B_{1}C_{1}D_{1}$ lying on the wrong side of $AD$, and so covering a
portion of the area of $ABCD$.

Taking $(0,\alpha)(-\ell,m)$ and $(\ell,m)$ for the coordinates of $A$, $B$, $D$ respectively, the formula of transformation is

$\frac{z_{1}-\alpha i}{z_{1}-\ell-mi}=\frac{\alpha}{m}\frac{z+\ell-mi}{z-\alpha},$ |

or what is the same thing

$z_{1}-\frac{\alpha\ell}{\alpha-m}=\frac{-\frac{m\alpha}{(\alpha-m)^{2}}\{\ell^% {2}+(\alpha-m)^{2}\}}{z+\frac{\alpha\ell}{\alpha-m}},$ |

leading at once to a geometrical construction, which is what I in fact made use in
drawing the figure: but I think one sees directly,
that $A$, $B$ being transformed
into $A_{1}$, $B_{1}$ respectively, a point $G$, to the right of $AB$ & indefinitely near it will
be transformed into a point $G_{1}$ to the left of $A_{1}B_{1}$ as shown in the figure. I have
ventured to trouble you with this difficulty & I shall be much obliged if you will
clear it up for me.^{3}^{3}endnote:
^{3}
Poincaré responded the following day; see
(§ 4-18-3). I remain dear Sir, yours very sincerely.

A. Cayley

ALS 3p. Private collection, Paris 75017.

Time-stamp: "14.11.2022 20:09"