4-18-2. Arthur Cayley to H. Poincaré

Cambridge 3rd 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.11endnote: 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).22endnote: 2 Poincaré (1882, § 7). Taking a symmetrical form ABCDABCD as shown in the figure, and applying to it the transformation which converts ABAB into DADA, I obtain a new form A1B1C1D1A_{1}B_{1}C_{1}D_{1} lying on the wrong side of ADAD, and so covering a portion of the area of ABCDABCD.

Taking (0,α)(-,m)(0,\alpha)(-\ell,m) and (,m)(\ell,m) for the coordinates of AA, BB, DD respectively, the formula of transformation is

z1-αiz1--mi=αmz+-miz-α,\frac{z_{1}-\alpha i}{z_{1}-\ell-mi}=\frac{\alpha}{m}\frac{z+\ell-mi}{z-\alpha},

or what is the same thing

z1-αα-m=-mα(α-m)2{2+(α-m)2}z+αα-m,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 AA, BB being transformed into A1A_{1}, B1B_{1} respectively, a point GG, to the right of ABAB & indefinitely near it will be transformed into a point G1G_{1} to the left of A1B1A_{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.33endnote: 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"



  • H. Poincaré (1882) Théorie des groupes fuchsiens. Acta mathematica 1, pp. 1–62. link1 Cited by: endnote 1, endnote 2.
  • H. Poincaré (1884) Notice sur les travaux scientifiques de M. Poincaré. Gauthier-Villars, Paris. link1 Cited by: endnote 1.