Volume 1 is an exposition of the techniques to solve the differential equations of mathematical physics, and contains an account of the applications of Mathieu functions to electrostatics.
Émile Mathieu was born into a family of minor civil servants.
His mother, Amélie Antoinette Aubertin, was from Metz while her brother, Pierre Aubertin, the uncle of our mathematician, had attended the École Polytechnique, was a colonel of artillery and the director of a foundry which made cannons.
He excelled at school, first in classical studies showing remarkable abilities in Latin and Greek compositions.
We mentioned above that his uncle Pierre Aubertin had studied at the École Polytechnique and he advised Émile on preparing himself for the entrance examinations which he took successfully in 1854.