A photo copy of the original correspondence is posted on the site of the University of Nancy in France. See Poincare Correspondence and navigate to the letters for Kowalevski. The following letter was translated from the French by Yvonne Coates ©2006 Yvonne Coates
For biograpical links MacTutor History of Mathematics Archive - Biographies are used.
Notes enclosed  were not in the text and are editorial comments added by the translator or by the creator of this site, Joan Spicci.
LETTER 1 FROM SOPHIE KOWALESKI TO HENRI POINCARE
25 October (1884) Sodertelje [Sweden - just south of Stockholm]
I thank you for your very welcome letter [lines]. I am very happy that you took the time to find the demonstration of M. Weierstrass's theorems, which I cite in my article; I don't believe that Mr. Weierstrass himself would have taken so much time [as you did] - I will send [you] with the greatest pleasure the example that I received from him [M. Weierstrass], but please [I wish to beg you to] excuse me if I can't do so within the next fortnight. At the moment I don't have in my hands the papers in question, and I also don't have much time [very many moments] I can dedicate to mathematics and [illegible, but it's something about a class]. I am going to begin to replace [or perhaps work them out] again. M. and Mme. Mittag-Leffler and I are living [all together] in a little house in the country near Stockholm, and we go into town twice a week for our class. [And] We have some other friends visiting for a while, the sort who won't leave us much time for serious work.
We have read with great interest your note in the Comptes Rendus concerning Fuch's article on [janissant?] differential equations with the same fundamental [illegible] as linear differential equations; but I don't feel you have, in your final point [conclusion], reached the same result [as Fuchs]. M. Leffler charged me to ask, in his name, if you might not wish to compile the interesting remarks you made in a more detailed manner, then send them to him for the Acta [Acta Mathematica]. [My French wasn't up to the final sentence of the paragraph, but I believe M. Leffler says that putting Poincare's remarks in the journal would give him the greatest pleasure.]
[illegible] well, Monsieur, give my compliments to Mme. Poincare and accept for yourself the expression of my highest consideration and deepest regard.
[signed] Sophie Kowalevski