4-18-2. Arthur Cayley to H. Poincaré
Cambridge 3rd March 1884
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 as shown in the figure, and applying to it the transformation which converts into , I obtain a new form lying on the wrong side of , and so covering a portion of the area of .
Taking and for the coordinates of , , respectively, the formula of transformation is
or what is the same thing
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 , being transformed into , respectively, a point , to the right of & indefinitely near it will be transformed into a point to the left of 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.
ALS 3p. Private collection, Paris 75017.
Time-stamp: "14.11.2022 20:09"