The author claims that the most radical break with prevailing conceptions was Poincaré's discovery of homoclinic points, which nowadays figure in studies of ``chaotic'' motions. The presence of a homoclinic point in a dynamical system complicates the orbit structure considerably and implies the existence of trajectories with quite unpredictable behaviour. Poincaré first encountered homoclinic points in 1889 in connection with his memoir for which he had been awarded a prize by the Swedish king Oscar II. The purpose of this article is to throw some light on the events connected with the memoir. The author is citing documents of the Mittag-Leffler Institute from which one can obtain a clear picture of what actually happened.