D'Alembert's strongest case was the one against Euler's paper on the precession of the equinoxes and the nutation of the earth's axis . An English translation of this article by Steven Jones is available by clicking here for the html version and here for the pdf version. The precession of the equinoxes is a phenomenon that has been known since classical times. The earth's axis is not in fact stationary and instead traces out a large circle with respect to the fixed stars, rather like a top spinning on an oblique axis. The period of the precession is about 26,000 years and it will significantly alter the location of the north celestial pole in the millennia to come. In 1748, the British Astronomer Royal James Bradley announced his discovery of another disturbance in the earth's axis of rotation, a nodding motion or "nutation" with an 18 year cycle. D'Alembert had set himself the task of explaining both phenomena in strictly mechanical terms, as a consequence of Newton's inverse-square law of gravitation. He eventually cracked the problem, and published his book-length solution  in the middle of 1749.
Euler had also been working on the problem of precession and nutation, but had not been able to solve it. He received a copy of d'Alembert's book late in the summer of 1749. In his "Observations" essay, sent to the Academy in June 1752, d'Alembert reported receiving a letter from Euler, dated January 3, 1750, in which Euler acknowledged receiving the book [6, p. 338]. Euler also said that he had not really been able to follow d'Alembert's argument, but that after he had read it, he saw the big picture and was able to give his own solution to the problem. It was this solution that Euler published in the Berlin Academy's 1749 volume. Euler's solution is certainly shorter (36 pages as opposed to d'Alembert's 184) and more comprehensible. Indeed, d'Alembert's mathematical writings were notorious for poor organization and impenetrability. More importantly, Euler's solution was far more general, and led him to an important paper the following year on general principles governing the motion of rigid bodies , a problem he had been working on since 1734 at least. So although Euler owed much to d'Alembert in his solution of the precession and equinox problem, there is also much in his paper  that is novel; all of this is explained in careful detail in a recent paper by Curtis Wilson .
Nevertheless, Euler ought to have acknowledged at the outset of the paper that he was only presenting an alternate solution to a problem that had already been solved by d'Alembert. It was a serious lapse of academic etiquette to have neglected this. In addition, the records show that Euler didn't actually present his results on the problem to the Berlin Academy until March 5, 1750, so it was ethically questionable for him to have inserted the article in the Academy's volume for 1749. In any case, Euler recognized the validity of D'Alembert's priority claim and inserted a brief notice  to this effect in the next volume of Academy's journal, published in translation here. In this notice, Euler acknowledges that he had written his paper only after he had read d'Alembert's book, and that he "makes no pretense to the glory that is due to he that first resolved this important question."
The majority of D'Alembert's "Observations" essay is occupied with his priority claim on the problem of precession and nutation. However, he also demanded that his priority be recognized for two other papers. Euler capitulated in one case but not in the other.