The first of Arbogast's two published works, his prize essay for the St. Petersburg Academy on 'the arbitrary functions introduced by the integration of differential equations which have more than two variables.' Even bolder than Euler in his conclusions, Arbogast 'showed that arbitrary functions may tolerate not only discontinuities in the Eulerian sense of the term, but also "combintaions of several portions of different curves or those drawn by the free movement of the hand", that is, discontinuities in the sense afterwards used by Augustin Cauchy' (DSB).