Note on the foregoing paper, “commutative ordinary differential operators," by J. L. Burchnall and J. W. Chaundy

Abstract

The argument of this paper map be regarded as centred round the ascertainment of two linear differential equations, those denoted by (P′ - p ) η -1 Ψ = 0, (Q′ — q ) η -1 Ψ = 0; these are, in fact, differential equations satisfied by a quotient of theta functions of several arguments when all of these arguments except one are taken constant, and the theory of the paper is in close connection with the classical method for approaching the definition of the theta function from the algebraical side. It would seem to be important to justify this assertion in precise terms, so as to make clear the general bearings of the results arrived at by the authors. The elementary algebraic method which they follow has in my opinion great interest, especially as it enables them to deal in part with the theory even when the curve has unknown multiple points; but one is not satisfied until the functions involved are defined by their behaviour and not by their algebraical form. I hope, therefore, that the writers will allow me to make the following remarks; and that I may be pardoned for explicit references to my own volume, ‘Abel’s Theorem,’ 1897. This was witten before the appearance of Weierstrass’ lectures (‘Werke,’ vol. 4, 1902; 600 quarto pages), and is very imperfect; but it is briefer in one respect than Weierstrass’ theory, by the use of the fundamental integral functions (Kronecker, Dedekind and Weber, Hensel). What there is of novelty in the following remarks relates mainly to these. The general ideas are expounded also in Clebsch and Gordan’s ‘Abelsche Functionen’ (1866)—to which, as to Weierstrass’ volumes, explicit references are given below. I consider in the first place a curve of which the multiple points are known and allowed for; but exemplify in examples how the theory can be applied when this is not so, the existence of a new multiple point being regarded as a particular case.