Commutative ordinary differential operators. II.—The identity Pn = Qm

In a paper previously communicated to the Society we developed the general theory of ordinary differential operators P, Q of interprime orders m, n, which satisfy the commutative identity PQ = QP. Such a pair of operators also satisfy a second identity f(P, Q) ≡ Pn — Qm + ... = 0, where, if we replace P, Q by two-way co-ordinates p, q, the equation f(p, q) = 0 defines a curve which is integral though not necessarily rational. The expression of P, Q depends upon parameters defined by equations of Abelian form involving transcendents of genus equal to that of the curve, which, if finite double points are absent, is g = ½ (m — 1) (n — 1). If the curve possesses finite double points, operators satisfying f (P, Q) = 0 can still be constructed by the method of C. O. 2, section IV, but the genus of the transcendents appearing will be less than ½ (m — 1) (n — 1). This number, however, in virtue of its position in the theory of lattice integers, is still of considerable importance.