II. On the Abelian system of differential equations, and their rational and integral algebraic integrals, with a discussion of the periodicity of Abelian functions

Before entering on the discussion of the Abelian system of differential equations, I treat of some general algebraic theorems having reference to the differences of various sets of “facients,” and give a wider definition to the term “source,” hitherto used to signify the source of a covariant, and treat of two operators, δ and ∆. I then show how, by forming what I call a “square-matrix,” all the conditions can be obtained which are fulfilled when a polynomial f(2) of the degree 2n in z is a perfect square. With regard to these conditions, I remark that any one of them being given all the others can be found by successive operations of the operator δ.