Appell was the President of the Société astronomique de France (SAF), the French astronomical society, from 1919 to 1921.

[1] His daughter Marguerite Appell (1883–1969), who married the mathematician Émile Borel, is known as a novelist under her pen-name Camille Marbo.

[3] He worked first on projective geometry in the line of Chasles, then on algebraic functions, differential equations, and complex analysis.

In 1926, with Professor Joseph-Marie Kampé de Fériet, he authored a treatise on generalized hypergeometric series.

He discovered a physical interpretation of the imaginary period of the doubly periodic function whose restriction to real arguments describes the motion of an ideal pendulum.