"4th. In elementary problems, the general term takes the form of a sine or cosine; the roots of the definite equation are either whole numbers, or real or irrational quantities, each... included between two definite limits. In more complex problems, the general term takes the form of a function given implicitly by means of a differential equation integrable or not. However it may be, the roots of the definite equation exist, they are real, infinite in number. This distinction of the parts of which the integral must be composed, is very important, since it shews... the form of the solution, and the necessary relation between the coefficients."