In 1820 Mossotti published two papers, one as an appendix to the ephemerides for 1821, concerning the shape and time of revolution of the sun, and one engineering paper on the movement of a spring. The inspiration of the latter, says Mossotti in the introduction to his paper, comes from a memoire by Jean and Jacques Bernoulli, Usage et Théorie d'une Machine qu'on peut nommer Instrument Ballistique. The goal of Mossotti's paper is to supply to some theoretical imperfections in the Bernoulli's memoire. His contributions to the problem must have been significant and very original, as some Italian scientists wrote to Carlini after reading Mossotti's paper asking for explanations. It may seem funny that an Astronomer should explain some of his Pupil's work, but we have to keep in mind that Mossotti was probably regarded as a sort of Enfant prodige in the Observatoire: some of his works were translated by famous German astronomers such as Enke; the level of his mathematical ability was such that he was able to correct some theoretical imperfections in the work of the Bernoulli brothers; Gauss, who published reviews of the main astronomical European publications of the time, usually devoted half a page to the Milan ephemerides compiled by Carlini, but when Mossotti published his appendices in 1816 and 1817 Gauss' review dedicated one paragraph to the actual ephemerides and two full pages to the appendices. Mossotti was an exceptional theoretician for his times, and some of the notation he established was to be taken as a standard by Italian scientists like Enrico Betti.
In 1821 he published a paper on the motion of an ellipsoidal aerostatic balloon, in which he analyses the possibility of success of such a project, and concludes that the theoretical possibility exists, but the available construction materials have to be perfected before attempting such a task. He also published an appendix for that year's ephemerides, called Opposizione di Giove nell'anno 1820 osservata al quadrante murale.
On 31 August 1822 Mossotti was accepted as a member of the Società Italiana delle Scienze residente in Modena, also known as ``The Society of 40" from the number of its members, in the Mathematics class, receiveing 9 votes, the relative majority over 33 voters (the closest competitor was Frullani, with 8 votes). Among the other members there were Legendre and Poisson.
Figure 4: Mossotti elected member of the Società dei 40 (M20).
In 1823 three of his letters to the Baron de Zach are published in his Correspondance, and the paper concerning the motion of water in canals ( Sul moto dell'acqua nei canali) is published in the Memorie della Società Italiana delle Scienze, Tomo XIX, Parte contenente le memorie di matematica; fascicolo secondo, pgg. 616-658. This paper begins with an interesting account of the history of hydraulics from Galileo to Mossotti's times, and goes on discussing general theory and particular examples.